Problems 10401 to 10500

Problem 10401


ODE	\[ \boxed {x^{1+n} y^{\prime }-x^{2 n} y^{2} a -y x^{n} b=c \,x^{m}+d} \]

program solution	\[ y = -\frac {x^{-n} \left (-2 \sqrt {c a}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}+1, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}+1, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right )\right ) x^{\frac {m}{2}}+\left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right )+\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right ) c_{3} \right ) \left (\sqrt {-4 a d +b^{2}+2 b n +n^{2}}+b +n \right )\right )}{2 a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right )+\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right ) c_{3} \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (\sqrt {a c}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}+1, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}+1, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right ) x^{\frac {m}{2}}-\frac {\left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right ) \left (\sqrt {-4 a d +b^{2}+2 b n +n^{2}}+b +n \right )}{2}\right ) x^{-n}}{a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b n +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right )} \]

Problem 10402


ODE	\[ \boxed {x \left (a \,x^{k}+b \right ) y^{\prime }-\alpha \,x^{n} y^{2}-\left (\beta -a n \,x^{k}\right ) y=\gamma \,x^{-n}} \]

program solution	\[ y = \frac {\left (a \,x^{k}+b \right )^{\frac {b n +\beta }{2 b k}} x^{-\frac {3 b n +\beta }{2 b}} \left (x^{\frac {-k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta \right ) \left (a \,x^{k}+b \right )^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a -b n -\beta }{2 b k}}-c_{3} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b}} \left (k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta \right ) \left (a \,x^{k}+b \right )^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a +b n +\beta }{2 b k}}\right )}{2 \alpha \left (\left (a \,x^{k}+b \right )^{-\frac {\sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} x^{\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} c_{3} +\left (a \,x^{k}+b \right )^{\frac {\sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}} x^{-\frac {k \sqrt {\frac {b^{2} n^{2}+2 \beta n b -4 \alpha \gamma +\beta ^{2}}{k^{2} a^{2}}}\, a}{2 b}}\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {x^{-n} \left (\tanh \left (\frac {\left (\left (-b n -\beta \right ) \ln \left (a \,x^{k}+b \right )+\left (\left (b n +\beta \right ) \ln \left (x \right )+c_{1} b \right ) k \right ) \sqrt {\left (b n +\beta \right )^{2} \left (n^{2} b^{2}+2 b \beta n -4 \gamma \alpha +\beta ^{2}\right )}}{2 k b \left (b n +\beta \right )^{2}}\right ) \sqrt {\left (b n +\beta \right )^{2} \left (n^{2} b^{2}+2 b \beta n -4 \gamma \alpha +\beta ^{2}\right )}+\left (b n +\beta \right )^{2}\right )}{2 \alpha \left (b n +\beta \right )} \]

Problem 10403


ODE	\[ \boxed {x^{2} \left (x^{n} a -1\right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (p \,x^{n}+q \right ) x y=-r \,x^{n}-s} \]

program solution	\[ \text {Expression too large to display} \] Warning, solution could not be veriﬁed

Maple solution	\[ \text {Expression too large to display} \]

Problem 10404


ODE	\[ \boxed {\left (x^{n} a +b \,x^{m}+c \right ) y^{\prime }-c y^{2}+b \,x^{m -1} y=a \,x^{-2+n}} \]

program solution	\[ y = -\frac {\left (x^{n} a +b \,x^{m}+c \right ) \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {x^{2} \left (x^{n} a +b \,x^{m}+c \right )^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+\left (b \left (m +1\right ) x^{m}+x^{n} n a \right ) x \left (x^{n} a +b \,x^{m}+c \right ) \textit {\_Y}^{\prime }\left (x \right )+a c \,x^{n} \textit {\_Y} \left (x \right )}{x^{2} \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{c \operatorname {DESol}\left (\left \{\frac {\left (x^{2 m} b^{2}+a^{2} x^{2 n}+2 b \left (x^{n} a +c \right ) x^{m}+2 x^{n} a c +c^{2}\right ) x^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) b^{2} x \left (m +1\right ) x^{2 m}+x^{2 n} \textit {\_Y}^{\prime }\left (x \right ) a^{2} n x +x \left (\left (a \left (1+m +n \right ) x^{n}+c \left (m +1\right )\right ) b \,x^{m}+x^{n} a c n \right ) \textit {\_Y}^{\prime }\left (x \right )+a c \,x^{n} \textit {\_Y} \left (x \right )}{x^{2} \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {Expression too large to display} \]

Problem 10405


ODE	\[ \boxed {\left (x^{n} a +b \,x^{m}+c \right ) y^{\prime }-a \,x^{-2+n} y^{2}-b \,x^{m -1} y=c} \]

program solution	\[ y = -\frac {x^{2-n} \left (x^{n} a +b \,x^{m}+c \right ) \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {x^{1+2 m} b^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+x^{1+2 n} a^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+2 a b \,x^{1+m +n} \textit {\_Y}^{\prime \prime }\left (x \right )+2 \left (a \,x^{1+n}+b \,x^{m +1}+\frac {c x}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+a c \,x^{-1+n} \textit {\_Y} \left (x \right )+2 \textit {\_Y}^{\prime }\left (x \right ) \left (\frac {a b \,x^{m +n} \left (m -n +3\right )}{2}+\frac {b^{2} \left (m -n +1\right ) x^{2 m}}{2}+a^{2} x^{2 n}-\frac {\left (-b \left (m -2 n +3\right ) x^{m}+\left (n -4\right ) a \,x^{n}+c \left (-2+n \right )\right ) c}{2}\right )}{x \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{a \operatorname {DESol}\left (\left \{\frac {x^{2} \left (x^{n} a +b \,x^{m}+c \right )^{2} \textit {\_Y}^{\prime \prime }\left (x \right )+2 \left (\frac {b \left (m -n +1\right ) x^{m}}{2}+x^{n} a -\frac {c \left (-2+n \right )}{2}\right ) \left (x^{n} a +b \,x^{m}+c \right ) x \textit {\_Y}^{\prime }\left (x \right )+a c \,x^{n} \textit {\_Y} \left (x \right )}{x^{2} \left (x^{n} a +b \,x^{m}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10406


ODE	\[ \boxed {\left (x^{n} a +b \,x^{m}+c \right ) y^{\prime }-\alpha \,x^{k} y^{2}-\beta \,x^{s} y=-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s}} \]

program solution

Maple solution	\[ y \left (x \right ) = \frac {-\alpha \left (\int \frac {x^{k} {\mathrm e}^{-\left (\int \frac {2 \alpha \,x^{k} \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x \right )}}{a \,x^{n}+b \,x^{m}+c}d x \right ) \lambda -\lambda c_{1} -{\mathrm e}^{-\left (\int \frac {2 \alpha \,x^{k} \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x \right )}}{c_{1} +\alpha \left (\int \frac {x^{k} {\mathrm e}^{-\left (\int \frac {2 \alpha \,x^{k} \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x \right )}}{a \,x^{n}+b \,x^{m}+c}d x \right )} \]

Problem 10407


ODE	\[ \boxed {\left (x^{n} a +b \,x^{m}+c \right ) \left (y^{\prime } x -y\right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right )=0} \]

program solution	\[ y = \frac {i \sqrt {-\frac {\lambda }{a^{2} x^{2 n}+2 b \,x^{m +n} a +2 x^{n} a c +x^{2 m} b^{2}+2 x^{m} b c +c^{2}}}\, \left (c_{3} {\mathrm e}^{i s \left (\int x^{k} \sqrt {-\frac {\lambda }{a^{2} x^{2 n}+2 b \,x^{m +n} a +2 x^{n} a c +x^{2 m} b^{2}+2 x^{m} b c +c^{2}}}d x \right )}-{\mathrm e}^{-i s \left (\int x^{k} \sqrt {-\frac {\lambda }{a^{2} x^{2 n}+2 b \,x^{m +n} a +2 x^{n} a c +x^{2 m} b^{2}+2 x^{m} b c +c^{2}}}d x \right )}\right ) x \left (x^{n} a +b \,x^{m}+c \right )}{c_{3} {\mathrm e}^{i s \left (\int x^{k} \sqrt {-\frac {\lambda }{a^{2} x^{2 n}+2 b \,x^{m +n} a +2 x^{n} a c +x^{2 m} b^{2}+2 x^{m} b c +c^{2}}}d x \right )}+{\mathrm e}^{-i s \left (\int x^{k} \sqrt {-\frac {\lambda }{a^{2} x^{2 n}+2 b \,x^{m +n} a +2 x^{n} a c +x^{2 m} b^{2}+2 x^{m} b c +c^{2}}}d x \right )}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \tanh \left (s \sqrt {\lambda }\, \left (\int \frac {x^{k}}{a \,x^{n}+b \,x^{m}+c}d x +c_{1} \right )\right ) x \sqrt {\lambda } \]

Problem 10408


ODE	\[ \boxed {-a y^{2}+y^{\prime }=b \,{\mathrm e}^{\lambda x}} \]

program solution	\[ y = \frac {\sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}} \left (\operatorname {BesselJ}\left (1, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right ) c_{3} +\operatorname {BesselY}\left (1, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )\right )}{\sqrt {a}\, \left (c_{3} \operatorname {BesselJ}\left (0, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )+\operatorname {BesselY}\left (0, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\sqrt {b}\, {\mathrm e}^{\frac {x \lambda }{2}} \left (\operatorname {BesselY}\left (1, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \lambda }{2}}}{\lambda }\right ) c_{1} +\operatorname {BesselJ}\left (1, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \lambda }{2}}}{\lambda }\right )\right )}{\sqrt {a}\, \left (c_{1} \operatorname {BesselY}\left (0, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \lambda }{2}}}{\lambda }\right )+\operatorname {BesselJ}\left (0, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \lambda }{2}}}{\lambda }\right )\right )} \]

Problem 10409


ODE	\[ \boxed {y^{\prime }-y^{2}=a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x}} \]

program solution	\[ y = \frac {{\mathrm e}^{\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }} \lambda +\operatorname {expIntegral}_{1}\left (-\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }\right ) {\mathrm e}^{\lambda x} a +{\mathrm e}^{\lambda x} c_{3} a}{c_{3} +\operatorname {expIntegral}_{1}\left (-\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{x \lambda } \operatorname {expIntegral}_{1}\left (-\frac {2 a \,{\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} a +{\mathrm e}^{\frac {2 a \,{\mathrm e}^{x \lambda }}{\lambda }} c_{1} \lambda +{\mathrm e}^{x \lambda } a}{\operatorname {expIntegral}_{1}\left (-\frac {2 a \,{\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} +1} \]

Problem 10410


ODE	\[ \boxed {y^{\prime }-\sigma y^{2}=a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x}} \]

program solution	\[ y = \frac {-c_{3} \operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \left (i \left (\sqrt {c}\, \sqrt {a}-\frac {b}{2}\right ) \sqrt {\sigma }+\frac {\lambda \sqrt {c}}{2}\right )+\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }-2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) \lambda \sqrt {c}+\left (-i c \sqrt {\sigma }\, {\mathrm e}^{\lambda x}-\frac {i b \sqrt {\sigma }}{2}+\frac {\lambda \sqrt {c}}{2}\right ) \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right )\right )}{\sqrt {c}\, \sigma \left (\operatorname {WhittakerM}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {i b \sqrt {\sigma }}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {c}\, \sqrt {\sigma }\, {\mathrm e}^{\lambda x}}{\lambda }\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b -2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) \left (i \left (\sqrt {c}\, \sqrt {a}-\frac {b}{2}\right ) \sqrt {\sigma }+\frac {\lambda \sqrt {c}}{2}\right )+\lambda c_{1} \operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b -2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) \sqrt {c}+\left (-i {\mathrm e}^{x \lambda } c \sqrt {\sigma }-\frac {i \sqrt {\sigma }\, b}{2}+\frac {\lambda \sqrt {c}}{2}\right ) \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right )\right )}{\sqrt {c}\, \sigma \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right )\right )} \]

Problem 10411


ODE	\[ \boxed {y^{\prime }-\sigma y^{2}-a y=b \,{\mathrm e}^{x}+c} \]

program solution	\[ y = \frac {2 c_{3} \sqrt {b}\, \sqrt {\sigma }\, {\mathrm e}^{\frac {x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {b}\, \sqrt {\sigma }\, {\mathrm e}^{\frac {x}{2}}\right )+2 \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {b}\, \sqrt {\sigma }\, {\mathrm e}^{\frac {x}{2}}\right ) \sqrt {b}\, \sqrt {\sigma }\, {\mathrm e}^{\frac {x}{2}}-\left (\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {b}\, \sqrt {\sigma }\, {\mathrm e}^{\frac {x}{2}}\right ) c_{3} +\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {b}\, \sqrt {\sigma }\, {\mathrm e}^{\frac {x}{2}}\right )\right ) \left (\sqrt {a^{2}-4 c \sigma }+a \right )}{2 \sigma \left (\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {b}\, \sqrt {\sigma }\, {\mathrm e}^{\frac {x}{2}}\right ) c_{3} +\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {b}\, \sqrt {\sigma }\, {\mathrm e}^{\frac {x}{2}}\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {-2 \sqrt {b}\, {\mathrm e}^{\frac {x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}+1, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) \sigma -2 \sqrt {b}\, {\mathrm e}^{\frac {x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}+1, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_{1} \sigma +\sqrt {\sigma }\, \left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_{1} +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )\right ) \left (\sqrt {a^{2}-4 \sigma c}+a \right )}{\sigma ^{\frac {3}{2}} \left (2 \operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_{1} +2 \operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )\right )} \]

Problem 10412


ODE	\[ \boxed {y^{\prime }-y^{2}-y b=a \left (\lambda -b \right ) {\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x}} \]

program solution	\[ y = \frac {\left (a \left (\left (\int {\mathrm e}^{\frac {b \lambda x +2 \,{\mathrm e}^{\lambda x} a}{\lambda }}d x \right ) c_{3} +1\right ) {\mathrm e}^{\frac {\lambda ^{2} x -{\mathrm e}^{\lambda x} a}{\lambda }}-c_{3} {\mathrm e}^{\frac {b \lambda x +{\mathrm e}^{\lambda x} a}{\lambda }}\right ) {\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }}}{\left (\int {\mathrm e}^{\frac {b \lambda x +2 \,{\mathrm e}^{\lambda x} a}{\lambda }}d x \right ) c_{3} +1} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{x \lambda } a \left (\int {\mathrm e}^{\frac {b \lambda x +2 \,{\mathrm e}^{x \lambda } a}{\lambda }}d x \right )+{\mathrm e}^{x \lambda } c_{1} a -{\mathrm e}^{\frac {b \lambda x +2 \,{\mathrm e}^{x \lambda } a}{\lambda }}}{\int {\mathrm e}^{\frac {b \lambda x +2 \,{\mathrm e}^{x \lambda } a}{\lambda }}d x +c_{1}} \]

Problem 10413


ODE	\[ \boxed {y^{\prime }-y^{2}-a \,{\mathrm e}^{\lambda x} y=-a \,{\mathrm e}^{\lambda x} b -b^{2}} \]

program solution	\[ y = \frac {\left (-2 \,{\mathrm e}^{\lambda x} a -3 b +2 \lambda \right ) \operatorname {WhittakerW}\left (\frac {-2 b +\lambda }{2 \lambda }, \frac {b}{\lambda }, \frac {a \,{\mathrm e}^{\lambda x}}{\lambda }\right )-2 \left ({\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{2 \lambda }} \left (\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }\right )^{\frac {2 b +\lambda }{2 \lambda }}-\frac {\operatorname {WhittakerM}\left (\frac {-2 b +\lambda }{2 \lambda }, \frac {b}{\lambda }, \frac {a \,{\mathrm e}^{\lambda x}}{\lambda }\right )}{2}\right ) b c_{3}}{\operatorname {WhittakerM}\left (\frac {-2 b +\lambda }{2 \lambda }, \frac {b}{\lambda }, \frac {a \,{\mathrm e}^{\lambda x}}{\lambda }\right ) c_{3} +\operatorname {WhittakerW}\left (\frac {-2 b +\lambda }{2 \lambda }, \frac {b}{\lambda }, \frac {a \,{\mathrm e}^{\lambda x}}{\lambda }\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-b \left (\int {\mathrm e}^{\frac {2 b \lambda x +{\mathrm e}^{x \lambda } a}{\lambda }}d x \right )+c_{1} b +{\mathrm e}^{\frac {2 b \lambda x +{\mathrm e}^{x \lambda } a}{\lambda }}}{-\left (\int {\mathrm e}^{\frac {2 b \lambda x +{\mathrm e}^{x \lambda } a}{\lambda }}d x \right )+c_{1}} \]

Problem 10414


ODE	\[ \boxed {y^{\prime }-y^{2}=a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4}} \]

program solution	\[ y = \frac {\lambda \left (-2 \Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1+n}{4+2 n}} \left (2+n \right )^{2} \left (b \,{\mathrm e}^{\lambda x}+{\mathrm e}^{2 \lambda x}\right ) \operatorname {BesselI}\left (\frac {n +3}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right )+\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (b^{2}-{\mathrm e}^{2 \lambda x}\right ) \left (2+n \right )-\csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) \pi \left (\left ({\mathrm e}^{\lambda x}+b \right ) \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}}-2 \,{\mathrm e}^{\lambda x} \operatorname {BesselI}\left (\frac {1+n}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (2+n \right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {n +3}{4+2 n}}\right ) c_{3} \right )}{2 \left ({\mathrm e}^{\lambda x}+b \right ) \left (-c_{3} \csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) \pi \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{\frac {1}{4+2 n}}+\Gamma \left (\frac {n +3}{2+n}\right )^{2} {\left (-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )}^{-\frac {1}{4+2 n}} \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a \left ({\mathrm e}^{\lambda x}+b \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \left (2+n \right ) \left ({\mathrm e}^{\lambda x}+b \right )\right )} \] Veriﬁed OK.

Maple solution	\[ \text {Expression too large to display} \]

Problem 10415


ODE	\[ \boxed {y^{\prime }-y^{2}=a \,{\mathrm e}^{8 \lambda x}+b \,{\mathrm e}^{6 \lambda x}+c \,{\mathrm e}^{4 \lambda x}-\lambda ^{2}} \]

program solution	\[ \text {Expression too large to display} \] Warning, solution could not be veriﬁed

Maple solution	\[ \text {Expression too large to display} \]

Problem 10416


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{k x} y^{2}=b \,{\mathrm e}^{s x}} \]

program solution	\[ y = -\frac {b \,{\mathrm e}^{s x} \left (c_{3} \operatorname {BesselJ}\left (\frac {s}{k +s}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right )+\operatorname {BesselY}\left (\frac {s}{k +s}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right )\right )}{\sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}} \operatorname {BesselJ}\left (\frac {k +2 s}{k +s}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right ) c_{3} +\operatorname {BesselY}\left (\frac {k +2 s}{k +s}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right ) \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}-s \left (c_{3} \operatorname {BesselJ}\left (\frac {s}{k +s}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right )+\operatorname {BesselY}\left (\frac {s}{k +s}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {b \,{\mathrm e}^{s x} \left (\operatorname {BesselY}\left (\frac {s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right )}{\operatorname {BesselJ}\left (\frac {2 s +k}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}+\sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}} \operatorname {BesselY}\left (\frac {2 s +k}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} -s \left (\operatorname {BesselY}\left (\frac {s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right )} \]

Problem 10417


ODE	\[ \boxed {y^{\prime }-b \,{\mathrm e}^{\mu x} y^{2}=a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x}} \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) {\mathrm e}^{-\mu x}}{b \operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10418


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} y^{2}-y b=c \,{\mathrm e}^{-\lambda x}} \]

program solution	\[ y = -\frac {\left (\left (b +\lambda -\sqrt {-4 c a +b^{2}+2 b \lambda +\lambda ^{2}}\right ) {\mathrm e}^{\frac {\left (b +\lambda -\sqrt {-4 c a +b^{2}+2 b \lambda +\lambda ^{2}}\right ) x}{2}}+c_{3} {\mathrm e}^{\frac {\left (b +\lambda +\sqrt {-4 c a +b^{2}+2 b \lambda +\lambda ^{2}}\right ) x}{2}} \left (b +\lambda +\sqrt {-4 c a +b^{2}+2 b \lambda +\lambda ^{2}}\right )\right ) {\mathrm e}^{-\lambda x}}{2 a \left (c_{3} {\mathrm e}^{\frac {\left (b +\lambda +\sqrt {-4 c a +b^{2}+2 b \lambda +\lambda ^{2}}\right ) x}{2}}+{\mathrm e}^{\frac {\left (b +\lambda -\sqrt {-4 c a +b^{2}+2 b \lambda +\lambda ^{2}}\right ) x}{2}}\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {\left (-\sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 \lambda b -\lambda ^{2}\right )}\, \tan \left (\frac {\left (\left (b +\lambda \right ) x +c_{1} \right ) \sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 \lambda b -\lambda ^{2}\right )}}{2 \left (b +\lambda \right )^{2}}\right )+\left (b +\lambda \right )^{2}\right ) {\mathrm e}^{-x \lambda }}{2 a \left (b +\lambda \right )} \]

Problem 10419


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\mu x} y^{2}-\lambda y=-a \,b^{2} {\mathrm e}^{\left (\mu +2 \lambda \right ) x}} \]

program solution	\[ y = \frac {b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} \left (c_{3} \cos \left (\frac {a b \sqrt {-{\mathrm e}^{2 \lambda x} {\mathrm e}^{2 \mu x}}}{\lambda +\mu }\right )-\sin \left (\frac {a b \sqrt {-{\mathrm e}^{2 \lambda x} {\mathrm e}^{2 \mu x}}}{\lambda +\mu }\right )\right )}{\sqrt {-{\mathrm e}^{2 \lambda x} {\mathrm e}^{2 \mu x}}\, \left (c_{3} \sin \left (\frac {a b \sqrt {-{\mathrm e}^{2 \lambda x} {\mathrm e}^{2 \mu x}}}{\lambda +\mu }\right )+\cos \left (\frac {a b \sqrt {-{\mathrm e}^{2 \lambda x} {\mathrm e}^{2 \mu x}}}{\lambda +\mu }\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {b \left (c_{1} \sinh \left (\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+\cosh \left (\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )\right ) {\mathrm e}^{x \lambda }}{c_{1} \cosh \left (\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+\sinh \left (\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )} \]

Problem 10420


ODE	\[ \boxed {y^{\prime }-{\mathrm e}^{\lambda x} y^{2}-a \,{\mathrm e}^{\mu x} y=a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x}} \]

program solution	\[ y = \frac {\lambda \left (\operatorname {hypergeom}\left (\left [\frac {\mu -\lambda }{\mu }\right ], \left [\frac {-\lambda +2 \mu }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right ) a \,{\mathrm e}^{\left (\mu -\lambda \right ) x}-c_{3} \left (\mu -\lambda \right )\right )}{\left (\mu -\lambda \right ) \left (c_{3} {\mathrm e}^{\lambda x}+\operatorname {hypergeom}\left (\left [-\frac {\lambda }{\mu }\right ], \left [\frac {\mu -\lambda }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\lambda \left (a c_{1} {\mathrm e}^{\left (\mu -\lambda \right ) x} \operatorname {hypergeom}\left (\left [\frac {\mu -\lambda }{\mu }\right ], \left [\frac {-\lambda +2 \mu }{\mu }\right ], \frac {a \,{\mathrm e}^{x \mu }}{\mu }\right )+\lambda -\mu \right )}{\left (\mu -\lambda \right ) \left (c_{1} \operatorname {hypergeom}\left (\left [-\frac {\lambda }{\mu }\right ], \left [\frac {\mu -\lambda }{\mu }\right ], \frac {a \,{\mathrm e}^{x \mu }}{\mu }\right )+{\mathrm e}^{x \lambda }\right )} \]

Problem 10421


ODE	\[ \boxed {y^{\prime }+{\mathrm e}^{\lambda x} y^{2} \lambda -a \,{\mathrm e}^{\mu x} y=-a \,{\mathrm e}^{\left (\mu -\lambda \right ) x}} \]

program solution	\[ y = \frac {-\operatorname {hypergeom}\left (\left [\frac {\mu -\lambda }{\mu }\right ], \left [\frac {-\lambda +2 \mu }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right ) a \,{\mathrm e}^{\left (\mu -\lambda \right ) x}+c_{3} \left (\mu -\lambda \right )}{\left (\mu -\lambda \right ) \left (c_{3} {\mathrm e}^{\lambda x}+\operatorname {hypergeom}\left (\left [-\frac {\lambda }{\mu }\right ], \left [\frac {\mu -\lambda }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {a c_{1} {\mathrm e}^{\left (\mu -\lambda \right ) x} \operatorname {hypergeom}\left (\left [\frac {\mu -\lambda }{\mu }\right ], \left [\frac {-\lambda +2 \mu }{\mu }\right ], \frac {a \,{\mathrm e}^{x \mu }}{\mu }\right )+\lambda -\mu }{\left (\lambda -\mu \right ) \left (c_{1} \operatorname {hypergeom}\left (\left [-\frac {\lambda }{\mu }\right ], \left [\frac {\mu -\lambda }{\mu }\right ], \frac {a \,{\mathrm e}^{x \mu }}{\mu }\right )+{\mathrm e}^{x \lambda }\right )} \]

Problem 10422


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\mu x} y^{2}-a b \,{\mathrm e}^{x \left (\lambda +\mu \right )} y=-b \lambda \,{\mathrm e}^{\lambda x}} \]

program solution	\[ y = -\frac {\left (6 \left (\mu +\frac {\lambda }{2}\right ) \left (-\frac {2 \left (\mu +\frac {\lambda }{2}\right ) \left (\lambda +\mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}}{3}+a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b \left (\frac {2 \lambda }{3}+\mu \right )\right ) \operatorname {WhittakerM}\left (\frac {\lambda +2 \mu }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+\left (\left (-\lambda ^{2}-3 \mu \lambda -2 \mu ^{2}\right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}+\left (a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+2 \left (\mu +\frac {\lambda }{2}\right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b +2 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \left (\mu +\frac {\lambda }{2}\right )\right ) b a \right ) \left (\lambda +\mu \right ) \operatorname {WhittakerM}\left (-\frac {\lambda }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+12 \left (\mu +\frac {\lambda }{2}\right )^{2} \left (\frac {2 \lambda }{3}+\mu \right ) {\mathrm e}^{-\frac {x \left (3 \lambda +2 \mu \right )}{2}} \left (\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )^{\frac {3 \lambda +4 \mu }{2 \lambda +2 \mu }}+a b c_{3} {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+x \left (\lambda +\mu \right )^{2}}{\lambda +\mu }}\right ) {\mathrm e}^{-\mu x}}{\left (4 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \left (\mu +\frac {\lambda }{2}\right )^{2} \operatorname {WhittakerM}\left (\frac {\lambda +2 \mu }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+\left (\lambda +\mu \right ) \left (\left (\lambda +2 \mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}+b a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}\right ) \operatorname {WhittakerM}\left (-\frac {\lambda }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+c_{3} {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }}\right ) a} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-6 \left (-\frac {2 \left (\lambda +\mu \right ) \left (\mu +\frac {\lambda }{2}\right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-4 \left (\lambda +\mu \right ) x \left (\mu +\frac {3 \lambda }{4}\right )}{2 \lambda +2 \mu }}}{3}+a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\lambda +\mu \right ) x \left (\mu +\frac {\lambda }{2}\right )}{2 \lambda +2 \mu }} b \left (\frac {2 \lambda }{3}+\mu \right )\right ) c_{1} \left (\mu +\frac {\lambda }{2}\right ) \operatorname {WhittakerM}\left (\frac {\lambda +2 \mu }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )-\left (\left (-\lambda ^{2}-3 \mu \lambda -2 \mu ^{2}\right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-4 \left (\lambda +\mu \right ) x \left (\mu +\frac {3 \lambda }{4}\right )}{2 \lambda +2 \mu }}+\left (\left (\lambda +2 \mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\lambda +\mu \right ) x \left (\mu +\frac {\lambda }{2}\right )}{2 \lambda +2 \mu }}+a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b \right ) b a \right ) \left (\lambda +\mu \right ) c_{1} \operatorname {WhittakerM}\left (-\frac {\lambda }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )-12 \,{\mathrm e}^{-\frac {\left (3 \lambda +4 \mu \right ) x}{2}} \left (\frac {2 \lambda }{3}+\mu \right ) c_{1} \left (\mu +\frac {\lambda }{2}\right )^{2} \left (\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )^{\frac {3 \lambda +4 \mu }{2 \lambda +2 \mu }}-b \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+x \lambda \left (\lambda +\mu \right )}{\lambda +\mu }} a}{\left (4 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\lambda +\mu \right ) x \left (\frac {3 \lambda }{2}+\mu \right )}{2 \lambda +2 \mu }} c_{1} \left (\mu +\frac {\lambda }{2}\right )^{2} \operatorname {WhittakerM}\left (\frac {\lambda +2 \mu }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+\left (\lambda +\mu \right ) c_{1} \left (\left (\lambda +2 \mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\lambda +\mu \right ) x \left (\frac {3 \lambda }{2}+\mu \right )}{2 \lambda +2 \mu }}+b a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}\right ) \operatorname {WhittakerM}\left (-\frac {\lambda }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }}\right ) a} \]

Problem 10423


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{k x} y^{2}-y b=c \,{\mathrm e}^{s x}+d \,{\mathrm e}^{-k x}} \]

program solution	\[ y = -\frac {{\mathrm e}^{-\frac {x \left (3 k +b \right )}{2}} \left (-2 \sqrt {c}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}+k +s}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}+k +s}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right )\right ) \sqrt {a}\, {\mathrm e}^{\frac {x \left (b +s +2 k \right )}{2}}+{\mathrm e}^{\frac {\left (b +k \right ) x}{2}} \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right )\right ) \left (\sqrt {-4 a d +b^{2}+2 b k +k^{2}}+b +k \right )\right )}{2 a \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (\sqrt {c}\, a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 k b +k^{2}}+s +k}{s +k}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 k b +k^{2}}+s +k}{s +k}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right ) {\mathrm e}^{\frac {x \left (s +k \right )}{2}}-\frac {\sqrt {a}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 k b +k^{2}}}{s +k}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 k b +k^{2}}}{s +k}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right ) \left (\sqrt {-4 a d +b^{2}+2 k b +k^{2}}+b +k \right )}{2}\right ) {\mathrm e}^{-k x}}{a^{\frac {3}{2}} \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 k b +k^{2}}}{s +k}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 k b +k^{2}}}{s +k}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right )} \]

Problem 10424


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} y^{2}-\left ({\mathrm e}^{x \left (\lambda +\mu \right )} b -\lambda \right ) y=c \,{\mathrm e}^{\mu x}} \]

program solution

Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-x \lambda } \left (\sqrt {4 a \,b^{2} c -b^{4}}\, \tan \left (\frac {\left ({\mathrm e}^{x \left (\lambda +\mu \right )} b +\left (\lambda +\mu \right ) c_{1} \right ) \sqrt {4 a \,b^{2} c -b^{4}}}{2 b^{2} \left (\lambda +\mu \right )}\right )-b^{2}\right )}{2 a b} \]

Problem 10425


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{k x} y^{2}-y b=c \,{\mathrm e}^{k n x}+d \,{\mathrm e}^{k \left (1+2 n \right ) x}} \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{{\mathrm e}^{2 k x \left (1+n \right )} \textit {\_Y} \left (x \right ) a d +{\mathrm e}^{k x \left (1+n \right )} \textit {\_Y} \left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right )+\left (-b -k \right ) \textit {\_Y}^{\prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) {\mathrm e}^{-k x}}{a \operatorname {DESol}\left (\left \{{\mathrm e}^{2 k x \left (1+n \right )} \textit {\_Y} \left (x \right ) a d +{\mathrm e}^{k x \left (1+n \right )} \textit {\_Y} \left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right )+\left (-b -k \right ) \textit {\_Y}^{\prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10426


ODE	\[ \boxed {y^{\prime }-{\mathrm e}^{\mu x} \left (y-b \,{\mathrm e}^{\lambda x}\right )^{2}=b \lambda \,{\mathrm e}^{\lambda x}} \]

program solution	\[ y = \frac {\left (\left ({\mathrm e}^{x \left (\lambda +\mu \right )} b -\frac {\mu \left (c_{3} +1\right )}{2}\right ) \cosh \left (\frac {\mu x}{2}\right )+\sinh \left (\frac {\mu x}{2}\right ) \left ({\mathrm e}^{x \left (\lambda +\mu \right )} b c_{3} -\frac {\mu \left (c_{3} +1\right )}{2}\right )\right ) {\mathrm e}^{-\mu x}}{c_{3} \sinh \left (\frac {\mu x}{2}\right )+\cosh \left (\frac {\mu x}{2}\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{x \left (\lambda +\mu \right )} c_{1} b \mu +b \,{\mathrm e}^{x \lambda }-c_{1} \mu ^{2}\right ) {\mathrm e}^{-x \mu }}{c_{1} \mu +{\mathrm e}^{-x \mu }} \]

Problem 10427


ODE	\[ \boxed {\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }-y^{2}-k \,{\mathrm e}^{\nu x} y=-m^{2}+k m \,{\mathrm e}^{\nu x}} \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )}{\operatorname {DESol}\left (\left \{\frac {\left (2 \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) \left (\lambda +\mu \right )\right ) b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}-k a \textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{x \left (\lambda +\nu \right )}-k \textit {\_Y}^{\prime }\left (x \right ) b \,{\mathrm e}^{x \left (\mu +\nu \right )}+a^{2} \left (\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \lambda x}+b^{2} \left (\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right ) {\mathrm e}^{2 \mu x}+2 \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\frac {c}{2}\right ) c \textit {\_Y}^{\prime \prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\lambda x} a c \lambda +\textit {\_Y}^{\prime }\left (x \right ) {\mathrm e}^{\mu x} b c \mu +k \left (\textit {\_Y} \left (x \right ) m -c \textit {\_Y}^{\prime }\left (x \right )\right ) {\mathrm e}^{\nu x}-\textit {\_Y} \left (x \right ) m^{2}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-m \left (\int \frac {{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )+c_{1} m -{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )}}{\int \frac {{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x -c_{1}} \]

Problem 10428


ODE	\[ \boxed {\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right ) \left (y^{\prime }-y^{2}\right )=-a \,\lambda ^{2} {\mathrm e}^{\lambda x}-b \,\mu ^{2} {\mathrm e}^{\mu x}} \]

program solution	\[ y = \frac {-c_{3} \left (a b \left (\lambda +\mu \right ) {\mathrm e}^{x \left (\lambda +\mu \right )}+a^{2} \lambda \,{\mathrm e}^{2 \lambda x}+{\mathrm e}^{2 \mu x} b^{2} \mu +c \left (a \lambda \,{\mathrm e}^{\lambda x}+b \mu \,{\mathrm e}^{\mu x}\right )\right ) \left (\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right )-a b \left (\lambda +\mu \right ) {\mathrm e}^{x \left (\lambda +\mu \right )}-a^{2} \lambda \,{\mathrm e}^{2 \lambda x}-c a \lambda \,{\mathrm e}^{\lambda x}-{\mathrm e}^{2 \mu x} b^{2} \mu -c b \mu \,{\mathrm e}^{\mu x}-c_{3}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2} \left (\left (\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right ) c_{3} +1\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (-a b \left (\lambda +\mu \right ) {\mathrm e}^{x \left (\lambda +\mu \right )}-a^{2} \lambda \,{\mathrm e}^{2 x \lambda }-{\mathrm e}^{2 x \mu } b^{2} \mu -c \left (a \lambda \,{\mathrm e}^{x \lambda }+b \mu \,{\mathrm e}^{x \mu }\right )\right ) \left (\int \frac {1}{\left ({\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c \right )^{2}}d x \right )-a b c_{1} \left (\lambda +\mu \right ) {\mathrm e}^{x \left (\lambda +\mu \right )}-{\mathrm e}^{2 x \lambda } c_{1} a^{2} \lambda -{\mathrm e}^{x \lambda } c_{1} a c \lambda -{\mathrm e}^{2 x \mu } c_{1} b^{2} \mu -{\mathrm e}^{x \mu } c_{1} b c \mu -1}{\left ({\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c \right )^{2} \left (c_{1} +\int \frac {1}{\left ({\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c \right )^{2}}d x \right )} \]

Problem 10429


ODE	\[ \boxed {y^{\prime }-y^{2}-a x \,{\mathrm e}^{\lambda x} y={\mathrm e}^{\lambda x} a} \]

program solution	\[ y = \frac {-\lambda ^{2} x -c_{3} \left (\int \frac {{\mathrm e}^{\frac {\left (\lambda x -1\right ) {\mathrm e}^{\lambda x} a}{\lambda ^{2}}}}{x^{2}}d x \right ) x -{\mathrm e}^{\frac {\left (\lambda x -1\right ) {\mathrm e}^{\lambda x} a}{\lambda ^{2}}} c_{3}}{x^{2} \left (\lambda ^{2}+\left (\int \frac {{\mathrm e}^{\frac {\left (\lambda x -1\right ) {\mathrm e}^{\lambda x} a}{\lambda ^{2}}}}{x^{2}}d x \right ) c_{3} \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-c_{1} \lambda ^{2} x +\left (\int \frac {{\mathrm e}^{\frac {{\mathrm e}^{x \lambda } a \left (x \lambda -1\right )}{\lambda ^{2}}}}{x^{2}}d x \right ) x +{\mathrm e}^{\frac {{\mathrm e}^{x \lambda } a \left (x \lambda -1\right )}{\lambda ^{2}}}}{x^{2} \left (c_{1} \lambda ^{2}-\left (\int \frac {{\mathrm e}^{\frac {{\mathrm e}^{x \lambda } a \left (x \lambda -1\right )}{\lambda ^{2}}}}{x^{2}}d x \right )\right )} \]

Problem 10430


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} y^{2}=b \,{\mathrm e}^{-\lambda x}} \]

program solution	\[ y = -\frac {\left (\left (\lambda -\sqrt {-4 a b +\lambda ^{2}}\right ) {\mathrm e}^{-\frac {\left (-\lambda +\sqrt {-4 a b +\lambda ^{2}}\right ) x}{2}}+c_{3} {\mathrm e}^{\frac {\left (\lambda +\sqrt {-4 a b +\lambda ^{2}}\right ) x}{2}} \left (\lambda +\sqrt {-4 a b +\lambda ^{2}}\right )\right ) {\mathrm e}^{-\lambda x}}{2 a \left (c_{3} {\mathrm e}^{\frac {\left (\lambda +\sqrt {-4 a b +\lambda ^{2}}\right ) x}{2}}+{\mathrm e}^{-\frac {\left (-\lambda +\sqrt {-4 a b +\lambda ^{2}}\right ) x}{2}}\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {\left (\lambda ^{2}-\tan \left (\frac {\sqrt {4 a b \,\lambda ^{2}-\lambda ^{4}}\, \left (x \lambda +c_{1} \right )}{2 \lambda ^{2}}\right ) \sqrt {4 a b \,\lambda ^{2}-\lambda ^{4}}\right ) {\mathrm e}^{-x \lambda }}{2 a \lambda } \]

Problem 10431


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} y^{2}=b n \,x^{-1+n}-a \,b^{2} {\mathrm e}^{\lambda x} x^{2 n}} \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{-x^{2 n} {\mathrm e}^{2 \lambda x} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{\lambda x} x^{-1+n} \textit {\_Y} \left (x \right ) a b n -\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) {\mathrm e}^{-\lambda x}}{a \operatorname {DESol}\left (\left \{-x^{2 n} {\mathrm e}^{2 \lambda x} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{\lambda x} x^{-1+n} \textit {\_Y} \left (x \right ) a b n -\textit {\_Y}^{\prime }\left (x \right ) \lambda +\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10432


ODE	\[ \boxed {y^{\prime }-{\mathrm e}^{\lambda x} y^{2}-a \,x^{n} y=a \lambda \,x^{n} {\mathrm e}^{-\lambda x}} \]

program solution	\[ y = -\frac {{\mathrm e}^{-\lambda x} \left (\left (\int {\mathrm e}^{\frac {a \,x^{1+n}-\lambda x \left (1+n \right )}{1+n}}d x \right ) \lambda -\lambda c_{3} +{\mathrm e}^{\frac {a \,x^{1+n}-\lambda x \left (1+n \right )}{1+n}}\right )}{\int {\mathrm e}^{\frac {x \left (x^{n} a -\lambda \left (1+n \right )\right )}{1+n}}d x -c_{3}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {{\mathrm e}^{-x \lambda } \left (\left (\int {\mathrm e}^{\frac {x^{n +1} a -x \lambda \left (n +1\right )}{n +1}}d x \right ) \lambda +\lambda c_{1} +{\mathrm e}^{\frac {x^{n +1} a -x \lambda \left (n +1\right )}{n +1}}\right )}{c_{1} +\int {\mathrm e}^{\frac {x \left (a \,x^{n}-\left (n +1\right ) \lambda \right )}{n +1}}d x} \]

Problem 10433


ODE	\[ \boxed {y^{\prime }+{\mathrm e}^{\lambda x} y^{2} \lambda -a \,x^{n} y \,{\mathrm e}^{\lambda x}=-x^{n} a} \]

program solution	\[ y = \frac {\left ({\mathrm e}^{\lambda x} \left (\int {\mathrm e}^{-\lambda x +a \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}d x \right ) \lambda +{\mathrm e}^{\lambda x} c_{3} \lambda ^{2}+{\mathrm e}^{a \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}\right ) {\mathrm e}^{-2 \lambda x}}{\lambda \left (\int {\mathrm e}^{-\lambda x +a \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}d x +\lambda c_{3} \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-x \lambda } \left (\int {\mathrm e}^{-x \lambda +a \left (\int {\mathrm e}^{x \lambda } x^{n}d x \right )}d x \right ) c_{1} \lambda +\lambda ^{2} {\mathrm e}^{-x \lambda }+c_{1} {\mathrm e}^{-2 x \lambda +a \left (\int {\mathrm e}^{x \lambda } x^{n}d x \right )}}{\lambda \left (\left (\int {\mathrm e}^{-x \lambda +a \left (\int {\mathrm e}^{x \lambda } x^{n}d x \right )}d x \right ) c_{1} +\lambda \right )} \]

Problem 10434


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} y^{2}+a b \,x^{n} {\mathrm e}^{\lambda x} y=b n \,x^{-1+n}} \]

program solution	\[ y = \frac {{\mathrm e}^{-\lambda x +a b \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )} \left (a b \,x^{n} {\mathrm e}^{\lambda x -a b \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )} \left (c_{3} +\lambda \left (\int {\mathrm e}^{\lambda x +a b \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}d x \right )\right )-\lambda \,{\mathrm e}^{\lambda x}\right )}{a \left (c_{3} +\lambda \left (\int {\mathrm e}^{\lambda x +a b \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}d x \right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {x^{n} \lambda \left (\int {\mathrm e}^{x \lambda +a b \left (\int {\mathrm e}^{x \lambda } x^{n}d x \right )}d x \right ) c_{1} a b +x^{n} a b -c_{1} \lambda \,{\mathrm e}^{a b \left (\int {\mathrm e}^{x \lambda } x^{n}d x \right )}}{a \left (\lambda \left (\int {\mathrm e}^{x \lambda +a b \left (\int {\mathrm e}^{x \lambda } x^{n}d x \right )}d x \right ) c_{1} +1\right )} \]

Problem 10435


ODE	\[ \boxed {y^{\prime }-a \,x^{n} y^{2}=b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}} \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-x^{1+2 n} {\mathrm e}^{2 \lambda x} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{\lambda x} x^{1+n} \textit {\_Y} \left (x \right ) a b \lambda +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-x^{1+2 n} {\mathrm e}^{2 \lambda x} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{\lambda x} x^{1+n} \textit {\_Y} \left (x \right ) a b \lambda +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10436


ODE	\[ \boxed {y^{\prime }-a \,x^{n} y^{2}-\lambda y=-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}} \]

program solution	\[ y = \frac {b \,{\mathrm e}^{\lambda x} \left (c_{3} \cosh \left (\frac {b \,x^{n} a \left (\left (\Gamma \left (n , -\lambda x \right ) n -\Gamma \left (1+n \right )\right ) \left (-\lambda x \right )^{-n}+{\mathrm e}^{\lambda x}\right )}{\lambda }\right )-\sinh \left (\frac {b \,x^{n} a \left (\left (\Gamma \left (n , -\lambda x \right ) n -\Gamma \left (1+n \right )\right ) \left (-\lambda x \right )^{-n}+{\mathrm e}^{\lambda x}\right )}{\lambda }\right )\right )}{-c_{3} \sinh \left (\frac {\left (-\lambda x \right )^{-n} x^{n} a b \left ({\mathrm e}^{\lambda x} \left (-\lambda x \right )^{n}+\Gamma \left (n , -\lambda x \right ) n -\Gamma \left (1+n \right )\right )}{\lambda }\right )+\cosh \left (\frac {\left (-\lambda x \right )^{-n} x^{n} a b \left ({\mathrm e}^{\lambda x} \left (-\lambda x \right )^{n}+\Gamma \left (n , -\lambda x \right ) n -\Gamma \left (1+n \right )\right )}{\lambda }\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \tanh \left (\frac {-a b \,x^{n} \left (n \Gamma \left (n , -x \lambda \right )-\Gamma \left (n +1\right )\right ) \left (-x \lambda \right )^{-n}-b a \,{\mathrm e}^{x \lambda } x^{n}+i \lambda c_{1}}{\lambda }\right ) b \,{\mathrm e}^{x \lambda } \]

Problem 10437


ODE	\[ \boxed {y^{\prime }-a \,x^{n} y^{2}+a b \,x^{n} {\mathrm e}^{\lambda x} y=b \lambda \,{\mathrm e}^{\lambda x}} \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {a b \,{\mathrm e}^{\lambda x} \left (\lambda \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime }\left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10438


ODE	\[ \boxed {y^{\prime }+\left (k +1\right ) x^{k} y^{2}-a \,x^{k +1} {\mathrm e}^{\lambda x} y=-{\mathrm e}^{\lambda x} a} \]

program solution	\[ y = \frac {\left (\int x^{-2 k -2} {\mathrm e}^{\int \left (a \,x^{k +1} {\mathrm e}^{\lambda x}+\frac {k}{x}\right )d x}d x +c_{3} \right ) \left (k +1\right ) x^{-k -1}+x^{-3 k -2} {\mathrm e}^{\int \left (a \,x^{k +1} {\mathrm e}^{\lambda x}+\frac {k}{x}\right )d x}}{\left (k +1\right ) \left (\int {\mathrm e}^{\int \frac {a \,x^{2+k} {\mathrm e}^{\lambda x}+k}{x}d x} x^{-2 k -2}d x +c_{3} \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {x^{-1-k} \left (x^{1+k} {\mathrm e}^{\int \frac {x^{1+k} {\mathrm e}^{x \lambda } a x -2 k -2}{x}d x}+\left (\int x^{k} {\mathrm e}^{a \left (\int x^{1+k} {\mathrm e}^{x \lambda }d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (1+k \right )}d x \right ) k +\int x^{k} {\mathrm e}^{a \left (\int x^{1+k} {\mathrm e}^{x \lambda }d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (1+k \right )}d x -c_{1} \right )}{\left (\int x^{k} {\mathrm e}^{a \left (\int x^{1+k} {\mathrm e}^{x \lambda }d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (1+k \right )}d x \right ) k +\int x^{k} {\mathrm e}^{a \left (\int x^{1+k} {\mathrm e}^{x \lambda }d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (1+k \right )}d x -c_{1}} \]

Problem 10439


ODE	\[ \boxed {y^{\prime }-a \,x^{n} y^{2}+a \,x^{n} \left (b \,{\mathrm e}^{\lambda x}+c \right ) y=c \,x^{n}} \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a c \,x^{1+2 n} \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) x +\left (x^{1+n} \left (b \,{\mathrm e}^{\lambda x}+c \right ) a -n \right ) \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {a c \,x^{1+2 n} \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) x +\left (x^{1+n} \left (b \,{\mathrm e}^{\lambda x}+c \right ) a -n \right ) \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10440


ODE	\[ \boxed {y^{\prime }-a \,x^{n} {\mathrm e}^{2 \lambda x} y^{2}-\left (b \,x^{n} {\mathrm e}^{\lambda x}-\lambda \right ) y=c \,x^{n}} \]

program solution	\[ y = \frac {{\mathrm e}^{-\lambda x} \left (\tan \left (\frac {\sqrt {4 a \,b^{2} c -b^{4}}\, \left (b n \left (-\lambda x \right )^{-n} \Gamma \left (n , -\lambda x \right ) x^{n}-x^{n} \left (-\lambda x \right )^{-n} \Gamma \left (1+n \right ) b +b \,x^{n} {\mathrm e}^{\lambda x}+\lambda c_{3} \right )}{2 b^{2} \lambda }\right ) \sqrt {4 a \,b^{2} c -b^{4}}-b^{2}\right )}{2 a b} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (\tan \left (\frac {\sqrt {4 a \,b^{2} c -b^{4}}\, \left (x^{n} \left (-x \lambda \right )^{-n} \Gamma \left (n , -x \lambda \right ) b n -x^{n} \left (-x \lambda \right )^{-n} \Gamma \left (n +1\right ) b +{\mathrm e}^{x \lambda } x^{n} b +\lambda c_{1} \right )}{2 b^{2} \lambda }\right ) \sqrt {4 a \,b^{2} c -b^{4}}-b^{2}\right ) {\mathrm e}^{-x \lambda }}{2 a b} \]

Problem 10441


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}=b n \,x^{-1+n}} \]

program solution	\[ y = \frac {\left (-\lambda \left (c_{3} +1\right ) {\mathrm e}^{-\lambda x}+2 \left (b \,x^{n}+c \right ) a \right ) \cosh \left (\frac {\lambda x}{2}\right )+2 \sinh \left (\frac {\lambda x}{2}\right ) \left (-\frac {\lambda \left (c_{3} +1\right ) {\mathrm e}^{-\lambda x}}{2}+\left (b \,x^{n}+c \right ) c_{3} a \right )}{2 \left (c_{3} \sinh \left (\frac {\lambda x}{2}\right )+\cosh \left (\frac {\lambda x}{2}\right )\right ) a} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {a c_{1} \lambda \left (b \,x^{n}+c \right ) {\mathrm e}^{x \lambda }+x^{n} a b -c_{1} \lambda ^{2}+a c}{\left (\lambda c_{1} {\mathrm e}^{x \lambda }+1\right ) a} \]

Problem 10442


ODE	\[ \boxed {y^{\prime } x -a \,{\mathrm e}^{\lambda x} y^{2}-k y=a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x}} \]

program solution	\[ y = \frac {b \,x^{k} \left (-c_{3} \cos \left (a b \,x^{k} \left (-\lambda x \right )^{-k} \left (\Gamma \left (k \right )-\Gamma \left (k , -\lambda x \right )\right )\right )+\sin \left (a b \,x^{k} \left (-\lambda x \right )^{-k} \left (\Gamma \left (k \right )-\Gamma \left (k , -\lambda x \right )\right )\right )\right )}{c_{3} \sin \left (a b \,x^{k} \left (-\lambda x \right )^{-k} \left (\Gamma \left (k \right )-\Gamma \left (k , -\lambda x \right )\right )\right )+\cos \left (a b \,x^{k} \left (-\lambda x \right )^{-k} \left (\Gamma \left (k \right )-\Gamma \left (k , -\lambda x \right )\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\tan \left (a b \,x^{k} \left (\Gamma \left (k , -x \lambda \right )-\Gamma \left (k \right )\right ) \left (-x \lambda \right )^{-k}+c_{1} \right ) b \,x^{k} \]

Problem 10443


ODE	\[ \boxed {y^{\prime } x -a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}-\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y={\mathrm e}^{\lambda x} c} \]

program solution	\[ y = -\frac {x^{-2 n} \left (\left (\lambda x +3 n \right ) {\mathrm e}^{-\lambda x}+b \,x^{n}\right )}{2 a} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {\left (\tan \left (\frac {\left (x^{n} b \left (\Gamma \left (n , -x \lambda \right )-\Gamma \left (n \right )\right ) \left (-x \lambda \right )^{-n}-c_{1} \right ) \sqrt {4 a \,b^{2} c -b^{4}}}{2 b^{2}}\right ) \sqrt {4 a \,b^{2} c -b^{4}}+b^{2}\right ) x^{-n}}{2 a b} \]

Problem 10444


ODE	\[ \boxed {y^{\prime }-y^{2}=2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} {\mathrm e}^{2 \lambda \,x^{2}}} \]

program solution	\[ y = -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{-{\mathrm e}^{2 \lambda \,x^{2}} \textit {\_Y} \left (x \right ) a^{2}+2 \,{\mathrm e}^{\lambda \,x^{2}} \textit {\_Y} \left (x \right ) a \lambda x +\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{-{\mathrm e}^{2 \lambda \,x^{2}} \textit {\_Y} \left (x \right ) a^{2}+2 \,{\mathrm e}^{\lambda \,x^{2}} \textit {\_Y} \left (x \right ) a \lambda x +\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10445


ODE	\[ \boxed {y^{\prime }-a \,{\mathrm e}^{-\lambda \,x^{2}} y^{2}-y \lambda x=a \,b^{2}} \]

program solution	\[ y = \frac {b \,{\mathrm e}^{\frac {\lambda \,x^{2}}{2}} \left (-c_{3} \cos \left (\frac {\sqrt {2}\, a b \sqrt {\pi }\, \operatorname {erf}\left (\frac {x \sqrt {2}\, \sqrt {\lambda }}{2}\right )}{2 \sqrt {\lambda }}\right )+\sin \left (\frac {\sqrt {2}\, a b \sqrt {\pi }\, \operatorname {erf}\left (\frac {x \sqrt {2}\, \sqrt {\lambda }}{2}\right )}{2 \sqrt {\lambda }}\right )\right )}{c_{3} \sin \left (\frac {\sqrt {2}\, a b \sqrt {\pi }\, \operatorname {erf}\left (\frac {x \sqrt {2}\, \sqrt {\lambda }}{2}\right )}{2 \sqrt {\lambda }}\right )+\cos \left (\frac {\sqrt {2}\, a b \sqrt {\pi }\, \operatorname {erf}\left (\frac {x \sqrt {2}\, \sqrt {\lambda }}{2}\right )}{2 \sqrt {\lambda }}\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \tan \left (\frac {a b \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {\lambda }\, x}{2}\right )-2 c_{1} \sqrt {\lambda }}{2 \sqrt {\lambda }}\right ) b \,{\mathrm e}^{\frac {x^{2} \lambda }{2}} \]

Problem 10446


ODE	\[ \boxed {y^{\prime }-a \,x^{n} y^{2}-y \lambda x=a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}}} \]

program solution	\[ y = \frac {b \,{\mathrm e}^{\frac {\lambda \,x^{2}}{2}} \left (-c_{3} \cos \left (a b \,x^{1+n} 2^{-\frac {1}{2}+\frac {n}{2}} \left (-\lambda \,x^{2}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (\Gamma \left (\frac {1}{2}+\frac {n}{2}\right )-\Gamma \left (\frac {1}{2}+\frac {n}{2}, -\frac {\lambda \,x^{2}}{2}\right )\right )\right )+\sin \left (a b \,x^{1+n} 2^{-\frac {1}{2}+\frac {n}{2}} \left (-\lambda \,x^{2}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (\Gamma \left (\frac {1}{2}+\frac {n}{2}\right )-\Gamma \left (\frac {1}{2}+\frac {n}{2}, -\frac {\lambda \,x^{2}}{2}\right )\right )\right )\right )}{c_{3} \sin \left (a b \,x^{1+n} 2^{-\frac {1}{2}+\frac {n}{2}} \left (-\lambda \,x^{2}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (\Gamma \left (\frac {1}{2}+\frac {n}{2}\right )-\Gamma \left (\frac {1}{2}+\frac {n}{2}, -\frac {\lambda \,x^{2}}{2}\right )\right )\right )+\cos \left (a b \,x^{1+n} 2^{-\frac {1}{2}+\frac {n}{2}} \left (-\lambda \,x^{2}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (\Gamma \left (\frac {1}{2}+\frac {n}{2}\right )-\Gamma \left (\frac {1}{2}+\frac {n}{2}, -\frac {\lambda \,x^{2}}{2}\right )\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\tan \left (-a b 2^{\frac {n}{2}-\frac {1}{2}} x^{n +1} \Gamma \left (\frac {n}{2}+\frac {1}{2}\right ) \left (-x^{2} \lambda \right )^{-\frac {n}{2}-\frac {1}{2}}+a b 2^{\frac {n}{2}-\frac {1}{2}} x^{n +1} \left (-x^{2} \lambda \right )^{-\frac {n}{2}-\frac {1}{2}} \Gamma \left (\frac {n}{2}+\frac {1}{2}, -\frac {x^{2} \lambda }{2}\right )+c_{1} \right ) b \,{\mathrm e}^{\frac {x^{2} \lambda }{2}} \]

Problem 10447


ODE	\[ \boxed {x^{4} \left (y^{\prime }-y^{2}\right )=a +b \,{\mathrm e}^{\frac {k}{x}}+c \,{\mathrm e}^{\frac {2 k}{x}}} \]

program solution	\[ y = \frac {-\frac {c_{3} \left (\left (-2 i \sqrt {a}-k \right ) \sqrt {c}+i b \right ) \operatorname {WhittakerM}\left (-\frac {-2 k \sqrt {c}+i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )}{2}-\operatorname {WhittakerW}\left (-\frac {-2 k \sqrt {c}+i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) k \sqrt {c}+\left (\operatorname {WhittakerM}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )\right ) \left (i c \,{\mathrm e}^{\frac {k}{x}}+\left (-\frac {k}{2}-x \right ) \sqrt {c}+\frac {i b}{2}\right )}{\sqrt {c}\, x^{2} \left (\operatorname {WhittakerM}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (\left (i \sqrt {a}+\frac {k}{2}\right ) \sqrt {c}-\frac {i b}{2}\right ) \operatorname {WhittakerM}\left (-\frac {i b -2 k \sqrt {c}}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )-c_{1} k \operatorname {WhittakerW}\left (-\frac {i b -2 k \sqrt {c}}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) \sqrt {c}+\left (\operatorname {WhittakerW}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )\right ) \left (i {\mathrm e}^{\frac {k}{x}} c +\left (-\frac {k}{2}-x \right ) \sqrt {c}+\frac {i b}{2}\right )}{\sqrt {c}\, x^{2} \left (\operatorname {WhittakerW}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i b}{2 k \sqrt {c}}, \frac {i \sqrt {a}}{k}, \frac {2 i \sqrt {c}\, {\mathrm e}^{\frac {k}{x}}}{k}\right )\right )} \]

Problem 10448


ODE	\[ \boxed {y^{\prime }-y^{2}=-a^{2}+a \lambda \sinh \left (\lambda x \right )-a^{2} \sinh \left (\lambda x \right )^{2}} \]

program solution	\[ y = \frac {\left (-2 a \sinh \left (\frac {i \pi }{4}+\frac {\lambda x}{2}\right ) \cosh \left (\lambda x \right )-\lambda \cosh \left (\frac {i \pi }{4}+\frac {\lambda x}{2}\right )\right ) \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\left (-2 a \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{3} +i \left (\operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{3} +\operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sinh \left (\frac {i \pi }{4}+\frac {\lambda x}{2}\right )\right ) \lambda \right ) \cosh \left (\lambda x \right )}{2 c_{3} \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )+2 \sinh \left (\frac {i \pi }{4}+\frac {\lambda x}{2}\right ) \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (-2 \sinh \left (\frac {i \pi }{4}+\frac {x \lambda }{2}\right ) \cosh \left (x \lambda \right ) c_{1} a -\cosh \left (\frac {i \pi }{4}+\frac {x \lambda }{2}\right ) c_{1} \lambda \right ) \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right )+\cosh \left (x \lambda \right ) \left (-2 \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right ) a +i \lambda \left (\operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right ) c_{1} \sinh \left (\frac {i \pi }{4}+\frac {x \lambda }{2}\right )+\operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right )\right )\right )}{2 \sinh \left (\frac {i \pi }{4}+\frac {x \lambda }{2}\right ) \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right ) c_{1} +2 \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right )} \]

Problem 10449


ODE	\[ \boxed {y^{\prime }-y^{2}-a \sinh \left (\beta x \right ) y=a b \sinh \left (\beta x \right )-b^{2}} \]

program solution	\[ y = \frac {-\left (\int {\mathrm e}^{\frac {-2 b \beta x +a \cosh \left (\beta x \right )}{\beta }}d x \right ) b \beta -{\mathrm e}^{\frac {-2 b \beta x +a \cosh \left (\beta x \right )}{\beta }} \beta -b c_{3}}{c_{3} +\beta \left (\int {\mathrm e}^{\frac {-2 b \beta x +a \cosh \left (\beta x \right )}{\beta }}d x \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {b \left (\int {\mathrm e}^{\frac {-2 b \beta x +a \cosh \left (x \beta \right )}{\beta }}d x \right )-c_{1} b +{\mathrm e}^{\frac {-2 b \beta x +a \cosh \left (x \beta \right )}{\beta }}}{-\left (\int {\mathrm e}^{\frac {-2 b \beta x +a \cosh \left (x \beta \right )}{\beta }}d x \right )+c_{1}} \]

Problem 10450


ODE	\[ \boxed {y^{\prime }-y^{2}-a x \sinh \left (b x \right )^{m} y=a \sinh \left (b x \right )^{m}} \]

program solution	\[ y = \frac {-{\mathrm e}^{\int \left (\sinh \left (b x \right )^{m} a x -\tanh \left (b x \right ) b \right )d x} \cosh \left (b x \right )-\left (\int \frac {{\mathrm e}^{\int \left (\sinh \left (b x \right )^{m} a x -\tanh \left (b x \right ) b \right )d x} \cosh \left (b x \right )}{x^{2}}d x \right ) x -c_{3} b x}{x^{2} \left (\int \frac {{\mathrm e}^{\int \left (\sinh \left (b x \right )^{m} a x -\tanh \left (b x \right ) b \right )d x} \cosh \left (b x \right )}{x^{2}}d x +b c_{3} \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-{\mathrm e}^{\int \frac {\sinh \left (b x \right )^{m} x^{2} a -2}{x}d x} x -\left (\int {\mathrm e}^{\int \frac {\sinh \left (b x \right )^{m} x^{2} a -2}{x}d x}d x \right )+c_{1}}{\left (-c_{1} +\int {\mathrm e}^{\int \frac {\sinh \left (b x \right )^{m} x^{2} a -2}{x}d x}d x \right ) x} \]

Problem 10451


ODE	\[ \boxed {y^{\prime }-\lambda \sinh \left (\lambda x \right ) y^{2}=-\lambda \sinh \left (\lambda x \right )^{3}} \]

program solution	\[ y = \frac {-2 \,{\mathrm e}^{\cosh \left (\lambda x \right )^{2}}+\cosh \left (\lambda x \right ) \sqrt {\pi }\, \left (c_{3} +\operatorname {erfi}\left (\cosh \left (\lambda x \right )\right )\right )}{\sqrt {\pi }\, \left (c_{3} +\operatorname {erfi}\left (\cosh \left (\lambda x \right )\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {2 \left ({\mathrm e}^{\frac {\cosh \left (2 x \lambda \right )}{2}+\frac {1}{2}} c_{1} -\frac {\cosh \left (x \lambda \right ) \sqrt {\pi }\, \left (\operatorname {erfi}\left (\cosh \left (x \lambda \right )\right ) c_{1} +1\right )}{2}\right )}{\sqrt {\pi }\, \left (\operatorname {erfi}\left (\cosh \left (x \lambda \right )\right ) c_{1} +1\right )} \]

Problem 10452


ODE	\[ \boxed {y^{\prime }-\left (a \sinh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}=-a \sinh \left (\lambda x \right )^{2}+\lambda -a} \]

program solution	\[ y = \frac {-2 \operatorname {csch}\left (\lambda x \right )^{2} {\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \lambda -2 \lambda \left (\int -{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \left (a -\operatorname {csch}\left (\lambda x \right )^{2} \lambda \right )d x \right ) \coth \left (\lambda x \right )+c_{3} \coth \left (\lambda x \right )}{-2 \lambda \left (\int -{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \left (a -\operatorname {csch}\left (\lambda x \right )^{2} \lambda \right )d x \right )+c_{3}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {2 \coth \left (x \lambda \right ) \lambda \left (\int -{\mathrm e}^{\frac {a \cosh \left (2 x \lambda \right )}{2 \lambda }} \left (a -\operatorname {csch}\left (x \lambda \right )^{2} \lambda \right )d x \right ) c_{1} +2 \operatorname {csch}\left (x \lambda \right )^{2} {\mathrm e}^{\frac {a \cosh \left (2 x \lambda \right )}{2 \lambda }} c_{1} \lambda -\coth \left (x \lambda \right )}{2 \lambda \left (\int -{\mathrm e}^{\frac {a \cosh \left (2 x \lambda \right )}{2 \lambda }} \left (a -\operatorname {csch}\left (x \lambda \right )^{2} \lambda \right )d x \right ) c_{1} -1} \]

Problem 10453


ODE	\[ \boxed {\left (a \sinh \left (\lambda x \right )+b \right ) y^{\prime }-y^{2}-c \sinh \left (\mu x \right ) y=-d^{2}+c d \sinh \left (\mu x \right )} \]

program solution

Maple solution	\[ y \left (x \right ) = \frac {-d \left (\int \frac {{\mathrm e}^{\frac {c \left (\int \frac {\sinh \left (x \mu \right )}{\sinh \left (x \lambda \right ) a +b}d x \right ) \lambda \sqrt {a^{2}+b^{2}}+4 d \,\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right )}{\lambda \sqrt {a^{2}+b^{2}}}}}{\sinh \left (x \lambda \right ) a +b}d x \right )+d c_{1} -{\mathrm e}^{\frac {c \left (\int \frac {\sinh \left (x \mu \right )}{\sinh \left (x \lambda \right ) a +b}d x \right ) \lambda \sqrt {a^{2}+b^{2}}+4 d \,\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right )}{\lambda \sqrt {a^{2}+b^{2}}}}}{\int \frac {{\mathrm e}^{\frac {c \left (\int \frac {\sinh \left (x \mu \right )}{\sinh \left (x \lambda \right ) a +b}d x \right ) \lambda \sqrt {a^{2}+b^{2}}+4 d \,\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right )}{\lambda \sqrt {a^{2}+b^{2}}}}}{\sinh \left (x \lambda \right ) a +b}d x -c_{1}} \]

Problem 10454


ODE	\[ \boxed {\left (a \sinh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )=-a \,\lambda ^{2} \sinh \left (\lambda x \right )} \]

program solution	\[ y = -\frac {4 \left (\left (\cosh \left (\frac {\lambda x}{2}\right )^{2}-\frac {1}{2}\right ) \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{3} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{3} -1\right ) a \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}+\sinh \left (\frac {\lambda x}{2}\right ) a b \cosh \left (\frac {\lambda x}{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) c_{3} \left (a^{2}+b^{2}\right )^{2}}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (2 \cosh \left (\frac {\lambda x}{2}\right ) a c_{3} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (a \sinh \left (\frac {\lambda x}{2}\right )+b \cosh \left (\frac {\lambda x}{2}\right )\right )+4 \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{3} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{3} -1\right ) \left (\sinh \left (\frac {\lambda x}{2}\right ) a \cosh \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {4 \left (\left (\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2}+\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4}-c_{1} \right ) a \left (\cosh \left (\frac {x \lambda }{2}\right )^{2}-\frac {1}{2}\right ) \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2}+b^{2}\right )^{2} \left (a^{2} \cosh \left (\frac {x \lambda }{2}\right )^{2}+a b \cosh \left (\frac {x \lambda }{2}\right ) \sinh \left (\frac {x \lambda }{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right )}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (2 a \cosh \left (\frac {x \lambda }{2}\right ) \left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (a \sinh \left (\frac {x \lambda }{2}\right )+b \cosh \left (\frac {x \lambda }{2}\right )\right )+4 \left (\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2}+\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4}-c_{1} \right ) \left (\sinh \left (\frac {x \lambda }{2}\right ) a \cosh \left (\frac {x \lambda }{2}\right )+\frac {b}{2}\right )\right )} \]

Problem 10455


ODE	\[ \boxed {y^{\prime }-\alpha y^{2}=\beta +\gamma \cosh \left (x \right )} \]

program solution	\[ y = -\frac {i \left (\operatorname {MathieuSPrime}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )+c_{3} \operatorname {MathieuCPrime}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )\right )}{2 \alpha \left (c_{3} \operatorname {MathieuC}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )+\operatorname {MathieuS}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {i \left (c_{1} \operatorname {MathieuSPrime}\left (-4 \alpha \beta , 2 \gamma \alpha , \frac {i x}{2}\right )+\operatorname {MathieuCPrime}\left (-4 \alpha \beta , 2 \gamma \alpha , \frac {i x}{2}\right )\right )}{2 \alpha \left (c_{1} \operatorname {MathieuS}\left (-4 \alpha \beta , 2 \gamma \alpha , \frac {i x}{2}\right )+\operatorname {MathieuC}\left (-4 \alpha \beta , 2 \gamma \alpha , \frac {i x}{2}\right )\right )} \]

Problem 10456


ODE	\[ \boxed {y^{\prime }-y^{2}-a \cosh \left (\beta x \right ) y=a b \cosh \left (\beta x \right )-b^{2}} \]

program solution	\[ y = \frac {-2 \,{\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} a \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) \cosh \left (\beta x \right )+\beta \operatorname {csch}\left (\frac {\beta x}{2}\right )^{2} \left ({\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} \operatorname {HeunDPrime}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )+\operatorname {HeunDPrime}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) c_{3} \right )}{2 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }}+2 c_{3} \operatorname {HeunD}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {b \left (\int {\mathrm e}^{\frac {-2 b \beta x +\sinh \left (x \beta \right ) a}{\beta }}d x \right )-c_{1} b +{\mathrm e}^{\frac {-2 b \beta x +\sinh \left (x \beta \right ) a}{\beta }}}{-\left (\int {\mathrm e}^{\frac {-2 b \beta x +\sinh \left (x \beta \right ) a}{\beta }}d x \right )+c_{1}} \]

Problem 10457


ODE	\[ \boxed {y^{\prime }-y^{2}-a x \cosh \left (b x \right )^{m} y=a \cosh \left (b x \right )^{m}} \]

program solution	\[ y = \frac {-\left (\int {\mathrm e}^{\int \frac {\cosh \left (b x \right )^{m} a \,x^{2}-2}{x}d x}d x \right ) c_{3} -1-x \,{\mathrm e}^{\int \frac {\cosh \left (b x \right )^{m} a \,x^{2}-2}{x}d x} c_{3}}{x \left (\left (\int {\mathrm e}^{\int \frac {\cosh \left (b x \right )^{m} a \,x^{2}-2}{x}d x}d x \right ) c_{3} +1\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-{\mathrm e}^{\int \frac {\cosh \left (b x \right )^{m} x^{2} a -2}{x}d x} x -\left (\int {\mathrm e}^{\int \frac {\cosh \left (b x \right )^{m} x^{2} a -2}{x}d x}d x \right )+c_{1}}{\left (-c_{1} +\int {\mathrm e}^{\int \frac {\cosh \left (b x \right )^{m} x^{2} a -2}{x}d x}d x \right ) x} \]

Problem 10458


ODE	\[ \boxed {y^{\prime }-\left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}=-a \cosh \left (\lambda x \right )^{2}+a +\lambda } \]

program solution	\[ y = \frac {-2 \,{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \operatorname {sech}\left (\lambda x \right )^{2} \lambda -2 \lambda \left (\int -{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \left (a -\operatorname {sech}\left (\lambda x \right )^{2} \lambda \right )d x \right ) \tanh \left (\lambda x \right )+c_{3} \tanh \left (\lambda x \right )}{-2 \lambda \left (\int -{\mathrm e}^{\frac {\cosh \left (2 \lambda x \right ) a}{2 \lambda }} \left (a -\operatorname {sech}\left (\lambda x \right )^{2} \lambda \right )d x \right )+c_{3}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {2 \tanh \left (x \lambda \right ) \lambda \left (\int -{\mathrm e}^{\frac {a \cosh \left (2 x \lambda \right )}{2 \lambda }} \left (a -\operatorname {sech}\left (x \lambda \right )^{2} \lambda \right )d x \right ) c_{1} +2 \operatorname {sech}\left (x \lambda \right )^{2} {\mathrm e}^{\frac {a \cosh \left (2 x \lambda \right )}{2 \lambda }} c_{1} \lambda -\tanh \left (x \lambda \right )}{2 \lambda \left (\int -{\mathrm e}^{\frac {a \cosh \left (2 x \lambda \right )}{2 \lambda }} \left (a -\operatorname {sech}\left (x \lambda \right )^{2} \lambda \right )d x \right ) c_{1} -1} \]

Problem 10459


ODE	\[ \boxed {2 y^{\prime }-\left (a -\lambda +a \cosh \left (\lambda x \right )\right ) y^{2}=a +\lambda -a \cosh \left (\lambda x \right )} \]

program solution	\[ y = \frac {\operatorname {sech}\left (\frac {\lambda x}{2}\right )^{2} \left (4 \,{\mathrm e}^{\frac {\cosh \left (\lambda x \right ) a}{\lambda }} \lambda +\lambda \left (\int {\mathrm e}^{\frac {\cosh \left (\lambda x \right ) a}{\lambda }} \left (-2 a +\operatorname {sech}\left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right ) \sinh \left (\lambda x \right )-2 c_{3} \sinh \left (\lambda x \right )\right )}{2 \lambda \left (\int {\mathrm e}^{\frac {\cosh \left (\lambda x \right ) a}{\lambda }} \left (-2 a +\operatorname {sech}\left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right )-4 c_{3}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\tanh \left (\frac {x \lambda }{2}\right ) \lambda \left (\int {\mathrm e}^{\frac {a \cosh \left (x \lambda \right )}{\lambda }} \left (-2 a +\operatorname {sech}\left (\frac {x \lambda }{2}\right )^{2} \lambda \right )d x \right ) c_{1} +2 \operatorname {sech}\left (\frac {x \lambda }{2}\right )^{2} {\mathrm e}^{\frac {a \cosh \left (x \lambda \right )}{\lambda }} c_{1} \lambda -2 \tanh \left (\frac {x \lambda }{2}\right )}{\lambda \left (\int {\mathrm e}^{\frac {a \cosh \left (x \lambda \right )}{\lambda }} \left (-2 a +\operatorname {sech}\left (\frac {x \lambda }{2}\right )^{2} \lambda \right )d x \right ) c_{1} -2} \]

Problem 10460


ODE	\[ \boxed {y^{\prime }-y^{2}=-\lambda ^{2}+a \cosh \left (\lambda x \right )^{n} \sinh \left (\lambda x \right )^{-n -4}} \]

program solution	\[ y = -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\sinh \left (\lambda x \right )^{-n -4} \textit {\_Y} \left (x \right ) \cosh \left (\lambda x \right )^{n} a -\textit {\_Y} \left (x \right ) \lambda ^{2}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\sinh \left (\lambda x \right )^{-n -4} \textit {\_Y} \left (x \right ) \cosh \left (\lambda x \right )^{n} a -\textit {\_Y} \left (x \right ) \lambda ^{2}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10461


ODE	\[ \boxed {y^{\prime }-y^{2} \sinh \left (\lambda x \right ) a=b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n}} \]

program solution	\[ y = -\frac {\left (-\sqrt {b}\, \left (\operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right ) c_{3} +\operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right ) \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \sqrt {a}+\lambda \left (c_{3} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right )\right ) \operatorname {sech}\left (\lambda x \right )}{\left (c_{3} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right ) a} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\operatorname {sech}\left (x \lambda \right ) \left (-\lambda \sqrt {a}\, \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )+\left (\operatorname {BesselY}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right ) a \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}\right )}{a^{\frac {3}{2}} \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )} \]

Problem 10462


ODE	\[ \boxed {y^{\prime }-\cosh \left (\lambda x \right ) y^{2} a=b \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right )^{n}} \]

program solution	\[ y = -\frac {\left (2+n \right ) \left (\Gamma \left (\frac {n +3}{2+n}\right )^{2} \left (-\frac {a b \sinh \left (\lambda x \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )^{\frac {1+n}{4+2 n}} \left (2+n \right ) \operatorname {BesselI}\left (\frac {n +3}{2+n}, 2 \sqrt {-\frac {a b \sinh \left (\lambda x \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right )-\operatorname {csch}\left (\lambda x \right ) \pi c_{3} \csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) \left (-\frac {a b \sinh \left (\lambda x \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )^{\frac {n +3}{4+2 n}} \operatorname {BesselI}\left (\frac {1+n}{2+n}, 2 \sqrt {-\frac {a b \sinh \left (\lambda x \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right )+\operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a b \sinh \left (\lambda x \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \Gamma \left (\frac {n +3}{2+n}\right )^{2} \left (-\frac {a b \sinh \left (\lambda x \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )^{-\frac {1}{4+2 n}}\right ) \lambda }{a \left (-\csc \left (\frac {\pi \left (n +3\right )}{2+n}\right ) c_{3} \operatorname {BesselI}\left (-\frac {1}{2+n}, 2 \sqrt {-\frac {a b \sinh \left (\lambda x \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \pi \left (-\frac {a b \sinh \left (\lambda x \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )^{\frac {1}{4+2 n}}+\sinh \left (\lambda x \right ) \operatorname {BesselI}\left (\frac {1}{2+n}, 2 \sqrt {-\frac {a b \sinh \left (\lambda x \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}}\right ) \Gamma \left (\frac {n +3}{2+n}\right )^{2} \left (-\frac {a b \sinh \left (\lambda x \right )^{2+n}}{\lambda ^{2} \left (2+n \right )^{2}}\right )^{-\frac {1}{4+2 n}} \left (2+n \right )\right )} \] Veriﬁed OK.

Maple solution	\[ \text {Expression too large to display} \]

Problem 10463


ODE	\[ \boxed {\left (a \cosh \left (\lambda x \right )+b \right ) y^{\prime }-y^{2}-c \cosh \left (\mu x \right ) y=-d^{2}+c d \cosh \left (\mu x \right )} \]

program solution

Maple solution	\[ y \left (x \right ) = \frac {-d \left (\int \frac {{\mathrm e}^{\frac {c \left (\int \frac {\cosh \left (x \mu \right )}{a \cosh \left (x \lambda \right )+b}d x \right ) \sqrt {a^{2}-b^{2}}\, \lambda -4 d \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \lambda }}}{a \cosh \left (x \lambda \right )+b}d x \right )+d c_{1} -{\mathrm e}^{\frac {c \left (\int \frac {\cosh \left (x \mu \right )}{a \cosh \left (x \lambda \right )+b}d x \right ) \sqrt {a^{2}-b^{2}}\, \lambda -4 d \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \lambda }}}{\int \frac {{\mathrm e}^{\frac {c \left (\int \frac {\cosh \left (x \mu \right )}{a \cosh \left (x \lambda \right )+b}d x \right ) \sqrt {a^{2}-b^{2}}\, \lambda -4 d \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \lambda }}}{a \cosh \left (x \lambda \right )+b}d x -c_{1}} \]

Problem 10464


ODE	\[ \boxed {\left (a \cosh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )=-a \,\lambda ^{2} \cosh \left (\lambda x \right )} \]

program solution	\[ y = \frac {\left (-2 b a \cosh \left (\frac {\lambda x}{2}\right ) \sinh \left (\frac {\lambda x}{2}\right ) c_{3} \arctan \left (\frac {\tanh \left (\frac {\lambda x}{2}\right ) \left (a -b \right )}{\sqrt {a^{2}-b^{2}}}\right ) \sqrt {a^{2}-b^{2}}+2 \sqrt {a^{2}-b^{2}}\, \sinh \left (\frac {\lambda x}{2}\right ) \cosh \left (\frac {\lambda x}{2}\right ) a +\left (a +b \right ) \left (a -b \right ) \left (a \cosh \left (\frac {\lambda x}{2}\right )^{2}-\frac {a}{2}-\frac {b}{2}\right ) c_{3} \right ) \lambda }{\sqrt {a^{2}-b^{2}}\, \left (2 b \left (a \cosh \left (\frac {\lambda x}{2}\right )^{2}-\frac {a}{2}+\frac {b}{2}\right ) c_{3} \arctan \left (\frac {\tanh \left (\frac {\lambda x}{2}\right ) \left (a -b \right )}{\sqrt {a^{2}-b^{2}}}\right )-\sqrt {a^{2}-b^{2}}\, \sinh \left (\frac {\lambda x}{2}\right ) \cosh \left (\frac {\lambda x}{2}\right ) c_{3} a -2 a \cosh \left (\frac {\lambda x}{2}\right )^{2}+a -b \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\lambda \left (-2 \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right ) \sqrt {a^{2}-b^{2}}\, a b \cosh \left (\frac {x \lambda }{2}\right ) \sinh \left (\frac {x \lambda }{2}\right )+2 \sqrt {a^{2}-b^{2}}\, c_{1} a \cosh \left (\frac {x \lambda }{2}\right ) \sinh \left (\frac {x \lambda }{2}\right )+\left (a +b \right ) \left (\cosh \left (\frac {x \lambda }{2}\right )^{2} a -\frac {a}{2}-\frac {b}{2}\right ) \left (a -b \right )\right )}{\sqrt {a^{2}-b^{2}}\, \left (2 \left (\cosh \left (\frac {x \lambda }{2}\right )^{2} a -\frac {a}{2}+\frac {b}{2}\right ) b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )-\sqrt {a^{2}-b^{2}}\, a \cosh \left (\frac {x \lambda }{2}\right ) \sinh \left (\frac {x \lambda }{2}\right )-2 c_{1} \left (\cosh \left (\frac {x \lambda }{2}\right )^{2} a -\frac {a}{2}+\frac {b}{2}\right )\right )} \]

Problem 10465


ODE	\[ \boxed {y^{\prime }-y^{2}=a \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}} \]

program solution	\[ y = \frac {\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) c_{3} \lambda +\operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) \lambda -\tanh \left (\lambda x \right ) \left (c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )\right ) \left (a +\lambda \right )}{c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right ) \lambda +\operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right ) c_{1} \lambda -\tanh \left (x \lambda \right ) \left (c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right )\right ) \left (a +\lambda \right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right )} \]

Problem 10466


ODE	\[ \boxed {y^{\prime }-y^{2}=3 a \lambda -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}} \]

program solution	\[ y = \frac {2 \operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right ) c_{3} \lambda +2 \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right ) \lambda -\tanh \left (\lambda x \right ) \left (c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )\right ) \left (a +\lambda \right )}{c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {2 \operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (x \lambda \right )\right ) \lambda +2 \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (x \lambda \right )\right ) c_{1} \lambda -\tanh \left (x \lambda \right ) \left (c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (x \lambda \right )\right )\right ) \left (a +\lambda \right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (x \lambda \right )\right )} \]

Problem 10467


ODE	\[ \boxed {y^{\prime }-y^{2}-a x \tanh \left (b x \right )^{m} y=a \tanh \left (b x \right )^{m}} \]

program solution	\[ y = \frac {-c_{3} \left (\int {\mathrm e}^{\int \frac {\tanh \left (b x \right )^{m} a \,x^{2}-2}{x}d x}d x \right )-1-x c_{3} {\mathrm e}^{\int \frac {\tanh \left (b x \right )^{m} a \,x^{2}-2}{x}d x}}{x \left (c_{3} \left (\int {\mathrm e}^{\int \frac {\tanh \left (b x \right )^{m} a \,x^{2}-2}{x}d x}d x \right )+1\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-{\mathrm e}^{\int \frac {a \tanh \left (b x \right )^{m} x^{2}-2}{x}d x} x -\left (\int {\mathrm e}^{\int \frac {a \tanh \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right )+c_{1}}{\left (-c_{1} +\int {\mathrm e}^{\int \frac {a \tanh \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right ) x} \]

Problem 10468


ODE	\[ \boxed {\left (a \tanh \left (\lambda x \right )+b \right ) y^{\prime }-y^{2}-c \tanh \left (\mu x \right ) y=-d^{2}+c d \tanh \left (\mu x \right )} \]

program solution

Maple solution	\[ y \left (x \right ) = \frac {-{\mathrm e}^{c \left (\int \frac {\tanh \left (x \mu \right )}{a \tanh \left (x \lambda \right )+b}d x \right )} \left (\tanh \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} \left (\tanh \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (a \tanh \left (x \lambda \right )+b \right )^{-\frac {2 a d}{\lambda \left (a^{2}-b^{2}\right )}}-d \left (\int \left (a \tanh \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\tanh \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\tanh \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\tanh \left (x \mu \right )}{a \tanh \left (x \lambda \right )+b}d x \right )}d x -c_{1} \right )}{\int \left (a \tanh \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\tanh \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\tanh \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\tanh \left (x \mu \right )}{a \tanh \left (x \lambda \right )+b}d x \right )}d x -c_{1}} \]

Problem 10469


ODE	\[ \boxed {y^{\prime }-y^{2}=a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}} \]

program solution	\[ y = \frac {\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right ) c_{3} \lambda +\operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right ) \lambda -\coth \left (\lambda x \right ) \left (c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )\right ) \left (a +\lambda \right )}{c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right ) \lambda +\operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right ) c_{1} \lambda -\coth \left (x \lambda \right ) \left (c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right )\right ) \left (a +\lambda \right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right )} \]

Problem 10470


ODE	\[ \boxed {y^{\prime }-y^{2}=3 a \lambda -\lambda ^{2}-a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}} \]

program solution	\[ y = \frac {2 \operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right ) c_{3} \lambda +2 \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right ) \lambda -\coth \left (\lambda x \right ) \left (c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )\right ) \left (a +\lambda \right )}{c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {2 \operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (x \lambda \right )\right ) \lambda +2 \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (x \lambda \right )\right ) c_{1} \lambda -\coth \left (x \lambda \right ) \left (c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (x \lambda \right )\right )\right ) \left (a +\lambda \right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (x \lambda \right )\right )} \]

Problem 10471


ODE	\[ \boxed {y^{\prime }-y^{2}-a x \coth \left (b x \right )^{m} y=a \coth \left (b x \right )^{m}} \]

program solution	\[ y = \frac {-\left (\int {\mathrm e}^{\int \frac {\coth \left (b x \right )^{m} a \,x^{2}-2}{x}d x}d x \right ) c_{3} -1-x \,{\mathrm e}^{\int \frac {\coth \left (b x \right )^{m} a \,x^{2}-2}{x}d x} c_{3}}{x \left (\left (\int {\mathrm e}^{\int \frac {\coth \left (b x \right )^{m} a \,x^{2}-2}{x}d x}d x \right ) c_{3} +1\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-{\mathrm e}^{\int \frac {a \coth \left (b x \right )^{m} x^{2}-2}{x}d x} x -\left (\int {\mathrm e}^{\int \frac {a \coth \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right )+c_{1}}{\left (-c_{1} +\int {\mathrm e}^{\int \frac {a \coth \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right ) x} \]

Problem 10472


ODE	\[ \boxed {\left (a \coth \left (\lambda x \right )+b \right ) y^{\prime }-y^{2}-c \coth \left (\mu x \right ) y=-d^{2}+c d \coth \left (\mu x \right )} \]

program solution

Maple solution	\[ y \left (x \right ) = \frac {-{\mathrm e}^{c \left (\int \frac {\coth \left (x \mu \right )}{a \coth \left (x \lambda \right )+b}d x \right )} \left (\coth \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} \left (\coth \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (a \coth \left (x \lambda \right )+b \right )^{-\frac {2 a d}{\lambda \left (a^{2}-b^{2}\right )}}-d \left (\int \left (a \coth \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\coth \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\coth \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\coth \left (x \mu \right )}{a \coth \left (x \lambda \right )+b}d x \right )}d x -c_{1} \right )}{\int \left (a \coth \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\coth \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\coth \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\coth \left (x \mu \right )}{a \coth \left (x \lambda \right )+b}d x \right )}d x -c_{1}} \]

Problem 10473


ODE	\[ \boxed {y^{\prime }-y^{2}=-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \coth \left (\lambda x \right )^{2} \lambda ^{2}} \]

program solution	\[ y = \frac {\operatorname {sech}\left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right ) \left (8 \cosh \left (\lambda x \right )^{5} \sinh \left (\lambda x \right )-8 \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{3}-2 \cosh \left (\lambda x \right )^{2} \ln \left (\coth \left (\lambda x \right )-1\right )+2 \cosh \left (\lambda x \right )^{2} \ln \left (\coth \left (\lambda x \right )+1\right )+2 \cosh \left (\lambda x \right )^{2} c_{3} -2 \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right )+\ln \left (\coth \left (\lambda x \right )-1\right )-\ln \left (\coth \left (\lambda x \right )+1\right )-c_{3} \right ) \lambda }{-4 \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{3}+2 \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right )-\ln \left (\coth \left (\lambda x \right )-1\right )+\ln \left (\coth \left (\lambda x \right )+1\right )+c_{3}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {2 \left (-\frac {1}{2}+c_{1} \left (-\cosh \left (x \lambda \right )^{2}+\frac {1}{2}\right ) \ln \left (\coth \left (x \lambda \right )-1\right )+c_{1} \left (\cosh \left (x \lambda \right )^{2}-\frac {1}{2}\right ) \ln \left (\coth \left (x \lambda \right )+1\right )+4 \cosh \left (x \lambda \right )^{5} c_{1} \sinh \left (x \lambda \right )-4 \cosh \left (x \lambda \right )^{3} c_{1} \sinh \left (x \lambda \right )-\sinh \left (x \lambda \right ) \cosh \left (x \lambda \right ) c_{1} +\cosh \left (x \lambda \right )^{2}\right ) \lambda \,\operatorname {csch}\left (x \lambda \right ) \operatorname {sech}\left (x \lambda \right )}{-4 \cosh \left (x \lambda \right )^{3} c_{1} \sinh \left (x \lambda \right )+2 \sinh \left (x \lambda \right ) \cosh \left (x \lambda \right ) c_{1} +\ln \left (\coth \left (x \lambda \right )+1\right ) c_{1} -\ln \left (\coth \left (x \lambda \right )-1\right ) c_{1} +1} \]

Problem 10474


ODE	\[ \boxed {y^{\prime }-y^{2}=-2 a b +a \lambda +b \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2}} \]

program solution	\[ y = \frac {-4 \left (-\frac {\lambda }{2}+b \right ) \coth \left (\lambda x \right )^{\frac {2 a +2 \lambda }{\lambda }} \lambda \operatorname {csch}\left (\lambda x \right )^{2} \operatorname {hypergeom}\left (\left [2, -\frac {2 b -3 \lambda }{2 \lambda }\right ], \left [\frac {2 a +5 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+\left (\left (\left (-3 a -3 b \right ) \lambda -2 a b \right ) \coth \left (\lambda x \right )^{\frac {2 a +2 \lambda }{\lambda }}-2 \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }} \left (-\frac {5 \,\operatorname {sech}\left (\lambda x \right ) \left (a +\frac {3 \lambda }{5}\right ) \lambda \,\operatorname {csch}\left (\lambda x \right )}{2}+a^{2} \tanh \left (\lambda x \right )\right )\right ) \operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+2 \left (b \coth \left (\lambda x \right )+a \tanh \left (\lambda x \right )\right ) \left (\frac {3 \lambda }{2}+a \right ) c_{3} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}}{\left (\operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right ) \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }}+c_{3} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}\right ) \left (2 a +3 \lambda \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-4 c_{1} \lambda \left (b -\frac {\lambda }{2}\right ) \coth \left (x \lambda \right )^{\frac {2 a +2 \lambda }{\lambda }} \operatorname {csch}\left (x \lambda \right )^{2} \operatorname {hypergeom}\left (\left [2, -\frac {2 b -3 \lambda }{2 \lambda }\right ], \left [\frac {2 a +5 \lambda }{2 \lambda }\right ], \coth \left (x \lambda \right )^{2}\right )-2 c_{1} \left (\left (\left (\frac {3 a}{2}+\frac {3 b}{2}\right ) \lambda +a b \right ) \coth \left (x \lambda \right )^{\frac {2 a +2 \lambda }{\lambda }}+\left (-\frac {5 \,\operatorname {sech}\left (x \lambda \right ) \lambda \left (a +\frac {3 \lambda }{5}\right ) \operatorname {csch}\left (x \lambda \right )}{2}+a^{2} \tanh \left (x \lambda \right )\right ) \coth \left (x \lambda \right )^{\frac {2 a +\lambda }{\lambda }}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (x \lambda \right )^{2}\right )+2 \left (a \tanh \left (x \lambda \right )+\coth \left (x \lambda \right ) b \right ) \left (a +\frac {3 \lambda }{2}\right ) \left (-\operatorname {csch}\left (x \lambda \right )^{2}\right )^{\frac {a +b}{\lambda }}}{\left (\operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (x \lambda \right )^{2}\right ) c_{1} \coth \left (x \lambda \right )^{\frac {2 a +\lambda }{\lambda }}+\left (-\operatorname {csch}\left (x \lambda \right )^{2}\right )^{\frac {a +b}{\lambda }}\right ) \left (2 a +3 \lambda \right )} \]

Problem 10475


ODE	\[ \boxed {y^{\prime }-a \ln \left (x \right )^{n} y^{2}=b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n}} \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}+a \textit {\_Y} \left (x \right ) \left (\ln \left (x \right )^{n} b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{2 n}\right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \ln \left (x \right )^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-a^{2} b^{2} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+2 n} x^{1+2 m}+a b m \,x^{m} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x \ln \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10476


ODE	\[ \boxed {y^{\prime } x -a y^{2}=b \ln \left (x \right )+c} \]

program solution	\[ y = \frac {\left (\operatorname {AiryAi}\left (1, -\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right ) c_{3} +\operatorname {AiryBi}\left (1, -\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right )\right ) \left (a b \right )^{\frac {1}{3}}}{a \left (c_{3} \operatorname {AiryAi}\left (-\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right )+\operatorname {AiryBi}\left (-\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (a b \right )^{\frac {1}{3}} \left (\operatorname {AiryBi}\left (1, -\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right ) c_{1} +\operatorname {AiryAi}\left (1, -\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right )\right )}{a \left (c_{1} \operatorname {AiryBi}\left (-\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right )+\operatorname {AiryAi}\left (-\frac {\left (a b \right )^{\frac {1}{3}} \left (b \ln \left (x \right )+c \right )}{b}\right )\right )} \]

Problem 10477


ODE	\[ \boxed {y^{\prime } x -a y^{2}=b \ln \left (x \right )^{k}+c \ln \left (x \right )^{2 k +2}} \]

program solution	\[ y = \frac {-\left (k +3\right ) c_{3} \left (i \left (k +1\right ) \sqrt {c}\, \sqrt {a}-a b \right ) \ln \left (x \right )^{k +1} \operatorname {hypergeom}\left (\left [\frac {\left (5+3 k \right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (4+2 k \right )}\right ], \left [\frac {3+2 k}{2+k}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \left (x \right )^{2+k}}{2+k}\right )+\left (k +1\right ) \left (-\ln \left (x \right )^{2+k} \left (i \sqrt {c}\, \left (k +3\right ) \sqrt {a}-a b \right ) \operatorname {hypergeom}\left (\left [\frac {\left (7+3 k \right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (4+2 k \right )}\right ], \left [\frac {5+2 k}{2+k}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \left (x \right )^{2+k}}{2+k}\right )+\left (k +3\right ) \left (\left (i \sqrt {c}\, \sqrt {a}\, \ln \left (x \right )^{2+k}-1\right ) \operatorname {hypergeom}\left (\left [\frac {\left (k +3\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (4+2 k \right )}\right ], \left [\frac {k +3}{2+k}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \left (x \right )^{2+k}}{2+k}\right )+i c_{3} \ln \left (x \right )^{k +1} \sqrt {a}\, \sqrt {c}\, \operatorname {hypergeom}\left (\left [\frac {\left (k +1\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (4+2 k \right )}\right ], \left [\frac {k +1}{2+k}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \left (x \right )^{2+k}}{2+k}\right )\right )\right )}{\left (k +3\right ) \left (k +1\right ) a \left (\operatorname {hypergeom}\left (\left [\frac {\left (k +1\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (4+2 k \right )}\right ], \left [\frac {k +1}{2+k}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \left (x \right )^{2+k}}{2+k}\right ) c_{3} +\operatorname {hypergeom}\left (\left [\frac {\left (k +3\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (4+2 k \right )}\right ], \left [\frac {k +3}{2+k}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, \ln \left (x \right )^{2+k}}{2+k}\right ) \ln \left (x \right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-\ln \left (x \right )^{1+k} \left (k +3\right ) \left (i \sqrt {c}\, \left (1+k \right ) \sqrt {a}-a b \right ) \operatorname {hypergeom}\left (\left [\frac {\left (3 k +5\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {2 k +3}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )+\left (-\left (i \left (k +3\right ) \sqrt {c}\, \sqrt {a}-a b \right ) \ln \left (x \right )^{k +2} c_{1} \operatorname {hypergeom}\left (\left [\frac {\left (3 k +7\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {2 k +5}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )+\left (k +3\right ) \left (\left (i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}-1\right ) c_{1} \operatorname {hypergeom}\left (\left [\frac {\left (k +3\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {k +3}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )+i \ln \left (x \right )^{1+k} \sqrt {a}\, \sqrt {c}\, \operatorname {hypergeom}\left (\left [\frac {\left (1+k \right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {1+k}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )\right )\right ) \left (1+k \right )}{\left (\operatorname {hypergeom}\left (\left [\frac {\left (k +3\right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {k +3}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right ) \ln \left (x \right ) c_{1} +\operatorname {hypergeom}\left (\left [\frac {\left (1+k \right ) \sqrt {c}+i \sqrt {a}\, b}{\sqrt {c}\, \left (2 k +4\right )}\right ], \left [\frac {1+k}{k +2}\right ], \frac {2 i \sqrt {a}\, \sqrt {c}\, \ln \left (x \right )^{k +2}}{k +2}\right )\right ) a \left (k +3\right ) \left (1+k \right )} \]

Problem 10478


ODE	\[ \boxed {y^{\prime } x -y^{2} x=-a^{2} x \ln \left (\beta x \right )^{2}+a} \]

program solution	\[ y = -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x -a \left (\ln \left (\beta x \right )^{2} a x -1\right ) \textit {\_Y} \left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x -a \left (\ln \left (\beta x \right )^{2} a x -1\right ) \textit {\_Y} \left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10479


ODE	\[ \boxed {y^{\prime } x -y^{2} x=-a^{2} x \ln \left (\beta x \right )^{2 k}+a k \ln \left (\beta x \right )^{k -1}} \]

program solution	\[ y = -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (a^{2} x \ln \left (\beta x \right )^{2 k}-a k \ln \left (\beta x \right )^{k -1}\right ) \textit {\_Y} \left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\frac {-\ln \left (\beta x \right )^{1+2 k} \textit {\_Y} \left (x \right ) a^{2} x +\ln \left (\beta x \right )^{k} \textit {\_Y} \left (x \right ) a k +\textit {\_Y}^{\prime \prime }\left (x \right ) \ln \left (\beta x \right ) x}{\ln \left (\beta x \right ) x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10480


ODE	\[ \boxed {y^{\prime } x -a \,x^{n} y^{2}=b -a \,b^{2} x^{n} \ln \left (x \right )^{2}} \]

program solution	\[ y = -\frac {x^{1-n} \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x^{2}-\ln \left (x \right )^{2} x^{2 n} \textit {\_Y} \left (x \right ) a^{2} b^{2}-\left (-1+n \right ) \textit {\_Y}^{\prime }\left (x \right ) x +x^{n} \textit {\_Y} \left (x \right ) a b}{x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{a \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x^{2}-\ln \left (x \right )^{2} x^{2 n} \textit {\_Y} \left (x \right ) a^{2} b^{2}-\left (-1+n \right ) \textit {\_Y}^{\prime }\left (x \right ) x +x^{n} \textit {\_Y} \left (x \right ) a b}{x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10481


ODE	\[ \boxed {x^{2} y^{\prime }-x^{2} y^{2}=a \ln \left (x \right )^{2}+b \ln \left (x \right )+c} \]

program solution	\[ y = \frac {\left (-\frac {b \ln \left (x \right ) \left (b \ln \left (x \right )-4 c +1\right ) a^{\frac {5}{2}}}{12}-\frac {b^{2} \left (b \ln \left (x \right )-c +\frac {1}{4}\right ) a^{\frac {3}{2}}}{12}+\frac {\ln \left (x \right )^{2} \left (c -\frac {1}{4}\right ) a^{\frac {7}{2}}}{3}-\frac {b^{4} \sqrt {a}}{48}-i \left (a \ln \left (x \right )+\frac {b}{2}\right )^{2} a^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {28 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {5}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )+\left (-\frac {a^{\frac {5}{2}} b}{4}+\left (-1-\frac {\ln \left (x \right )}{2}\right ) a^{\frac {7}{2}}+i \left (a \ln \left (x \right )+\frac {b}{2}\right )^{2} a^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {12 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )+\frac {c_{3} \left (\left (\ln \left (x \right ) \left (c -\frac {1}{4}\right ) a^{\frac {5}{2}}-\frac {b \left (b \ln \left (x \right )-2 c +\frac {1}{2}\right ) a^{\frac {3}{2}}}{4}-\frac {b^{3} \sqrt {a}}{8}-i \left (a \ln \left (x \right )+\frac {b}{2}\right ) a^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {20 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )+\left (i \ln \left (x \right ) a^{3}+\frac {i a^{2} b}{2}-\frac {a^{\frac {5}{2}}}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {4 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )\right )}{2}}{a^{\frac {5}{2}} \left (\left (a \ln \left (x \right )+\frac {b}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {12 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )+\frac {c_{3} \operatorname {hypergeom}\left (\left [\frac {4 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )}{2}\right ) x} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (-\frac {b \ln \left (x \right ) \left (b \ln \left (x \right )-4 c +1\right ) a^{\frac {5}{2}}}{12}-\frac {\left (b \ln \left (x \right )-c +\frac {1}{4}\right ) b^{2} a^{\frac {3}{2}}}{12}+\frac {\left (c -\frac {1}{4}\right ) \ln \left (x \right )^{2} a^{\frac {7}{2}}}{3}-\frac {\sqrt {a}\, b^{4}}{48}-i \left (a \ln \left (x \right )+\frac {b}{2}\right )^{2} a^{2}\right ) c_{1} \operatorname {hypergeom}\left (\left [\frac {28 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {5}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )+c_{1} \left (-\frac {a^{\frac {5}{2}} b}{4}+\left (-1-\frac {\ln \left (x \right )}{2}\right ) a^{\frac {7}{2}}+i \left (a \ln \left (x \right )+\frac {b}{2}\right )^{2} a^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {12 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )+\frac {\left (\left (c -\frac {1}{4}\right ) \ln \left (x \right ) a^{\frac {5}{2}}-\frac {b \left (b \ln \left (x \right )-2 c +\frac {1}{2}\right ) a^{\frac {3}{2}}}{4}-\frac {\sqrt {a}\, b^{3}}{8}-i \left (a \ln \left (x \right )+\frac {b}{2}\right ) a^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {20 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )}{2}+\frac {\left (i a^{3} \ln \left (x \right )+\frac {i a^{2} b}{2}-\frac {a^{\frac {5}{2}}}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {4 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )}{2}}{a^{\frac {5}{2}} x \left (c_{1} \left (a \ln \left (x \right )+\frac {b}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {12 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )+\frac {\operatorname {hypergeom}\left (\left [\frac {4 a^{\frac {3}{2}}+i \left (4 c -1\right ) a -i b^{2}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a \ln \left (x \right )+b \right )^{2}}{4 a^{\frac {3}{2}}}\right )}{2}\right )} \]

Problem 10482


ODE	\[ \boxed {x^{2} y^{\prime }-x^{2} y^{2}=a \left (b \ln \left (x \right )+c \right )^{n}+\frac {1}{4}} \]

program solution	\[ y = -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {4 \left (b \ln \left (x \right )+c \right )^{n} \textit {\_Y} \left (x \right ) a +4 \textit {\_Y}^{\prime \prime }\left (x \right ) x^{2}+\textit {\_Y} \left (x \right )}{4 x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\frac {4 \left (b \ln \left (x \right )+c \right )^{n} \textit {\_Y} \left (x \right ) a +4 \textit {\_Y}^{\prime \prime }\left (x \right ) x^{2}+\textit {\_Y} \left (x \right )}{4 x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10483


ODE	\[ \boxed {x^{2} \ln \left (x a \right ) \left (y^{\prime }-y^{2}\right )=1} \]

program solution	\[ y = \frac {-\operatorname {expIntegral}_{1}\left (-\ln \left (x a \right )\right )+c_{3}}{x \left (\operatorname {expIntegral}_{1}\left (-\ln \left (x a \right )\right ) \ln \left (x a \right )-c_{3} \ln \left (x a \right )+x a \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-c_{1} \operatorname {expIntegral}_{1}\left (-\ln \left (a x \right )\right )+1}{x \left (\left (c_{1} \operatorname {expIntegral}_{1}\left (-\ln \left (a x \right )\right )-1\right ) \ln \left (a x \right )+c_{1} a x \right )} \]

Problem 10484


ODE	\[ \boxed {y^{\prime }-y^{2}-a \ln \left (\beta x \right ) y=-a b \ln \left (\beta x \right )-b^{2}} \]

program solution	\[ y = \frac {\left (\int \left (\beta x \right )^{x a} {\mathrm e}^{-x \left (a -2 b \right )}d x +c_{3} \right ) b -\left (\beta x \right )^{x a} {\mathrm e}^{-x \left (a -2 b \right )}}{\int \left (\beta x \right )^{x a} {\mathrm e}^{-x \left (a -2 b \right )}d x +c_{3}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (\int \left (x \beta \right )^{a x} {\mathrm e}^{-\left (a -2 b \right ) x}d x -c_{1} \right ) b -\left (x \beta \right )^{a x} {\mathrm e}^{-\left (a -2 b \right ) x}}{\int \left (x \beta \right )^{a x} {\mathrm e}^{-\left (a -2 b \right ) x}d x -c_{1}} \]

Problem 10485


ODE	\[ \boxed {y^{\prime }-y^{2}-a x \ln \left (b x \right )^{m} y=a \ln \left (b x \right )^{m}} \]

program solution	\[ y = \frac {-\left (\int {\mathrm e}^{\int \frac {\ln \left (b x \right )^{m} a \,x^{2}-2}{x}d x}d x \right ) c_{3} -1-x \,{\mathrm e}^{\int \frac {\ln \left (b x \right )^{m} a \,x^{2}-2}{x}d x} c_{3}}{x \left (\left (\int {\mathrm e}^{\int \frac {\ln \left (b x \right )^{m} a \,x^{2}-2}{x}d x}d x \right ) c_{3} +1\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-{\mathrm e}^{\int \frac {a \ln \left (b x \right )^{m} x^{2}-2}{x}d x} x -\left (\int {\mathrm e}^{\int \frac {a \ln \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right )+c_{1}}{\left (-c_{1} +\int {\mathrm e}^{\int \frac {a \ln \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right ) x} \]

Problem 10486


ODE	\[ \boxed {y^{\prime }-a \,x^{n} y^{2}+a b \,x^{1+n} \ln \left (x \right ) y=b \ln \left (x \right )+b} \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {b a \textit {\_Y} \left (x \right ) \left (1+\ln \left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x +\textit {\_Y}^{\prime }\left (x \right ) \left (b \ln \left (x \right ) x^{2+n} a -n \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {b a \textit {\_Y} \left (x \right ) \left (1+\ln \left (x \right )\right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x +\textit {\_Y}^{\prime }\left (x \right ) \left (b \ln \left (x \right ) x^{2+n} a -n \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 10487


ODE	\[ \boxed {y^{\prime }+\left (1+n \right ) x^{n} y^{2}-a \,x^{1+n} \ln \left (x \right )^{m} y=-a \ln \left (x \right )^{m}} \]

program solution	\[ y = \frac {\left (\int x^{-2 n -2} {\mathrm e}^{\int \left (a \,x^{1+n} \ln \left (x \right )^{m}+\frac {n}{x}\right )d x}d x +c_{3} \right ) \left (1+n \right ) x^{-1-n}+x^{-2-3 n} {\mathrm e}^{\int \left (a \,x^{1+n} \ln \left (x \right )^{m}+\frac {n}{x}\right )d x}}{\left (1+n \right ) \left (\int {\mathrm e}^{\int \frac {\ln \left (x \right )^{m} x^{2+n} a +n}{x}d x} x^{-2 n -2}d x +c_{3} \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {x^{-n -1} \left (x^{n +1} {\mathrm e}^{\int \frac {a \,x^{n +1} \ln \left (x \right )^{m} x -2 n -2}{x}d x}+\left (\int x^{n} {\mathrm e}^{a \left (\int x^{n +1} \ln \left (x \right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (n +1\right )}d x \right ) n +\int x^{n} {\mathrm e}^{a \left (\int x^{n +1} \ln \left (x \right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (n +1\right )}d x -c_{1} \right )}{\left (\int x^{n} {\mathrm e}^{a \left (\int x^{n +1} \ln \left (x \right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (n +1\right )}d x \right ) n +\int x^{n} {\mathrm e}^{a \left (\int x^{n +1} \ln \left (x \right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (n +1\right )}d x -c_{1}} \]

Problem 10488


ODE	\[ \boxed {y^{\prime }-a \ln \left (x \right )^{n} y+a b x \ln \left (x \right )^{1+n} y=b \ln \left (x \right )+b} \]

program solution	\[ \int _{}^{x}\left (a y \left (-1+b \textit {\_a} \ln \left (\textit {\_a} \right )\right ) \ln \left (\textit {\_a} \right )^{n}-b \left (1+\ln \left (\textit {\_a} \right )\right )\right ) {\mathrm e}^{a \left (\int \ln \left (\textit {\_a} \right )^{n} \left (-1+b \textit {\_a} \ln \left (\textit {\_a} \right )\right )d \textit {\_a} \right )}d \textit {\_a} +\left (-{\mathrm e}^{a \left (\int _{}^{x}\ln \left (\textit {\_a} \right )^{n} \left (\ln \left (\textit {\_a} \right ) \textit {\_a} b -1\right )d \textit {\_a} \right )}+{\mathrm e}^{a \left (\int \ln \left (x \right )^{n} \left (-1+b x \ln \left (x \right )\right )d x \right )}\right ) y = c_{1} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \left (b \left (\int {\mathrm e}^{a \left (\int \ln \left (x \right )^{n} \left (-1+\ln \left (x \right ) b x \right )d x \right )} \left (\ln \left (x \right )+1\right )d x \right )+c_{1} \right ) {\mathrm e}^{-a \left (\int \ln \left (x \right )^{n} \left (-1+\ln \left (x \right ) b x \right )d x \right )} \]

Problem 10489


ODE	\[ \boxed {y^{\prime }-a \ln \left (x \right )^{k} \left (y-b \,x^{n}-c \right )^{2}=b n \,x^{-1+n}} \]

program solution	\[ y = \frac {\ln \left (x \right )^{-k} \left (\left (\left (x^{1+n} b +c x \right ) \left (-\ln \left (x \right )\right )^{k}-\left (b \,x^{n}+c \right ) \left (-\Gamma \left (k , -\ln \left (x \right )\right ) k +\Gamma \left (k +1\right )\right )\right ) a \ln \left (x \right )^{\frac {3 k}{2}}+\left (-\ln \left (x \right )\right )^{k} \ln \left (x \right )^{\frac {k}{2}} \left (x^{n} c_{3} a b +c_{3} c a -1\right )\right )}{\left (\left (x \left (-\ln \left (x \right )\right )^{k}+\Gamma \left (k , -\ln \left (x \right )\right ) k -\Gamma \left (k +1\right )\right ) \ln \left (x \right )^{\frac {k}{2}}+\ln \left (x \right )^{-\frac {k}{2}} c_{3} \left (-\ln \left (x \right )\right )^{k}\right ) a} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = b \,x^{n}+c +\frac {1}{c_{1} -a \left (\int \ln \left (x \right )^{k}d x \right )} \]

Problem 10490


ODE	\[ \boxed {y^{\prime }-a \ln \left (x \right )^{n} y^{2}-b \ln \left (x \right )^{m} y=b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n}} \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-\ln \left (x \right ) \ln \left (x \right )^{2 n} \textit {\_Y} \left (x \right ) a^{2} c^{2} x +\ln \left (x \right ) \ln \left (x \right )^{m +n} \textit {\_Y} \left (x \right ) a b c x -\textit {\_Y}^{\prime }\left (x \right ) \ln \left (x \right )^{m +1} b x +\textit {\_Y}^{\prime \prime }\left (x \right ) x \ln \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \ln \left (x \right )^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-x \,a^{2} c^{2} \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+2 n}+a b c x \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+m +n}-\textit {\_Y}^{\prime }\left (x \right ) \ln \left (x \right )^{m +1} b x +\textit {\_Y}^{\prime \prime }\left (x \right ) x \ln \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{x \ln \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-c a \left (\int \ln \left (x \right )^{n} {\mathrm e}^{-\left (\int \left (2 \ln \left (x \right )^{n} a c -\ln \left (x \right )^{m} b \right )d x \right )}d x \right )-c_{1} c -{\mathrm e}^{-\left (\int \left (2 \ln \left (x \right )^{n} a c -\ln \left (x \right )^{m} b \right )d x \right )}}{c_{1} +a \left (\int \ln \left (x \right )^{n} {\mathrm e}^{-\left (\int \left (2 \ln \left (x \right )^{n} a c -\ln \left (x \right )^{m} b \right )d x \right )}d x \right )} \]

Problem 10491


ODE	\[ \boxed {y^{\prime } x -\left (a y+b \ln \left (x \right )\right )^{2}=0} \]

program solution	\[ y = \frac {-c_{3} \left (a \ln \left (x \right ) b +\sqrt {-a b}\right ) x^{2 \sqrt {-a b}}-a \ln \left (x \right ) b +\sqrt {-a b}}{a^{2} \left (1+x^{2 \sqrt {-a b}} c_{3} \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-\ln \left (x \right ) a b +\tan \left (\left (\ln \left (x \right )+c_{1} \right ) \sqrt {a b}\right ) \sqrt {a b}}{a^{2}} \]

Problem 10492


ODE	\[ \boxed {y^{\prime } x -a \ln \left (\lambda x \right )^{m} y^{2}-k y=a \,b^{2} x^{2 k} \ln \left (\lambda x \right )^{m}} \]

program solution	\[ y = -\frac {i b \,x^{k} \left (c_{3} {\mathrm e}^{2 i a b \left (\int x^{k -1} \ln \left (\lambda x \right )^{m}d x \right )}-1\right )}{c_{3} {\mathrm e}^{2 i a b \left (\int x^{k -1} \ln \left (\lambda x \right )^{m}d x \right )}+1} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\tan \left (-a b \left (\int x^{-1+k} \ln \left (x \lambda \right )^{m}d x \right )+c_{1} \right ) b \,x^{k} \]

Problem 10493


ODE	\[ \boxed {y^{\prime } x -a \,x^{n} \left (y+b \ln \left (x \right )\right )^{2}=-b} \]

program solution	\[ y = -\frac {\left (a b \,x^{\frac {n^{2}+b a \,x^{n}}{n}} \ln \left (x \right )+x^{\frac {a b \,x^{n}}{n}} \left (\ln \left (x \right ) c_{3} a b +n \right )\right ) x^{-\frac {a b \,x^{n}}{n}}}{a \left (x^{n}+c_{3} \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -b \ln \left (x \right )+\frac {n}{c_{1} n -a \,x^{n}} \]

Problem 10494


ODE	\[ \boxed {y^{\prime } x -a \,x^{2 n} \ln \left (x \right ) y^{2}-\left (b \,x^{n} \ln \left (x \right )-n \right ) y=c \ln \left (x \right )} \]

program solution	\[ y = -\frac {x^{-2 n} \left (1+\ln \left (x \right )^{2} x^{n} b +3 n \ln \left (x \right )\right )}{2 a \ln \left (x \right )^{2}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (\tan \left (\frac {\left (b \left (n \ln \left (x \right )-1\right ) x^{n}+c_{1} n^{2}\right ) \sqrt {4 a \,b^{2} c -b^{4}}}{2 b^{2} n^{2}}\right ) \sqrt {4 a \,b^{2} c -b^{4}}-b^{2}\right ) x^{-n}}{2 a b} \]

Problem 10495


ODE	\[ \boxed {x^{2} y^{\prime }-y^{2} a^{2} x^{2}+x y=b^{2} \ln \left (x \right )^{n}} \]

program solution	\[ y = \frac {\sqrt {b^{2} a^{2}}\, \operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) \ln \left (x \right )^{1+\frac {n}{2}} c_{3} +\sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )-\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )-\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{3}}{\ln \left (x \right ) x \,a^{2} \left (\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right )+\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {b^{2} a^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{2+n}\right ) c_{3} \right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\ln \left (x \right )^{\frac {n}{2}+1} \operatorname {BesselY}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a^{2} b^{2}}\, c_{1} +\operatorname {BesselJ}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} -\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )}{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) a^{2} x \ln \left (x \right )} \]

Problem 10496


ODE	\[ \boxed {\left (a \ln \left (x \right )+b \right ) y^{\prime }-y^{2}-c \ln \left (x \right )^{n} y=-\lambda ^{2}+\lambda c \ln \left (x \right )^{n}} \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-a c x \ln \left (x \right )^{1+n} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) \left (a \ln \left (x \right )+b \right )^{2} x +\left (-b \ln \left (x \right )^{n} c x +\left (a \ln \left (x \right )+b \right ) a \right ) \textit {\_Y}^{\prime }\left (x \right )+\lambda \left (\ln \left (x \right )^{n} c -\lambda \right ) \textit {\_Y} \left (x \right ) x}{\left (a \ln \left (x \right )+b \right )^{2} x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \left (a \ln \left (x \right )+b \right )}{\operatorname {DESol}\left (\left \{\frac {-a c x \ln \left (x \right )^{1+n} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) \left (a \ln \left (x \right )+b \right )^{2} x +\left (-b \ln \left (x \right )^{n} c x +\left (a \ln \left (x \right )+b \right ) a \right ) \textit {\_Y}^{\prime }\left (x \right )+\lambda \left (\ln \left (x \right )^{n} c -\lambda \right ) \textit {\_Y} \left (x \right ) x}{\left (a \ln \left (x \right )+b \right )^{2} x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-\left (\int \frac {{\mathrm e}^{\int \frac {\ln \left (x \right )^{n} c -2 \lambda }{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x \right ) \lambda -\lambda c_{1} -{\mathrm e}^{\int \frac {\ln \left (x \right )^{n} c -2 \lambda }{a \ln \left (x \right )+b}d x}}{c_{1} +\int \frac {{\mathrm e}^{\int \frac {\ln \left (x \right )^{n} c -2 \lambda }{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x} \]

Problem 10497


ODE	\[ \boxed {\left (a \ln \left (x \right )+b \right ) y^{\prime }-\ln \left (x \right )^{n} y^{2}-y c=-\lambda ^{2} \ln \left (x \right )^{n}+c \lambda } \]

program solution	\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-x \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+2 n} \lambda ^{2}+c x \lambda \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+n}+\left (a \ln \left (x \right )+b \right ) \left (x \ln \left (x \right ) \left (a \ln \left (x \right )+b \right ) \textit {\_Y}^{\prime \prime }\left (x \right )-\left (\left (a \left (-1+n \right )+c x \right ) \ln \left (x \right )+b n \right ) \textit {\_Y}^{\prime }\left (x \right )\right )}{x \ln \left (x \right ) \left (a \ln \left (x \right )+b \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \ln \left (x \right )^{-n} \left (a \ln \left (x \right )+b \right )}{\operatorname {DESol}\left (\left \{\frac {-x \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+2 n} \lambda ^{2}+c x \lambda \textit {\_Y} \left (x \right ) \ln \left (x \right )^{1+n}+\left (a \ln \left (x \right )+b \right ) \left (x \ln \left (x \right ) \left (a \ln \left (x \right )+b \right ) \textit {\_Y}^{\prime \prime }\left (x \right )-\left (\left (a \left (-1+n \right )+c x \right ) \ln \left (x \right )+b n \right ) \textit {\_Y}^{\prime }\left (x \right )\right )}{x \ln \left (x \right ) \left (a \ln \left (x \right )+b \right )^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {-\lambda c_{1} -\left (\int \frac {\ln \left (x \right )^{n} {\mathrm e}^{-\left (\int \frac {2 \ln \left (x \right )^{n} \lambda -c}{a \ln \left (x \right )+b}d x \right )}}{a \ln \left (x \right )+b}d x \right ) \lambda -{\mathrm e}^{-\left (\int \frac {2 \ln \left (x \right )^{n} \lambda -c}{a \ln \left (x \right )+b}d x \right )}}{c_{1} +\int \frac {\ln \left (x \right )^{n} {\mathrm e}^{-\left (\int \frac {2 \ln \left (x \right )^{n} \lambda -c}{a \ln \left (x \right )+b}d x \right )}}{a \ln \left (x \right )+b}d x} \]

Problem 10498


ODE	\[ \boxed {y^{\prime }-\alpha y^{2}=\beta +\gamma \sin \left (\lambda x \right )} \]

program solution	\[ y = -\frac {\lambda \left (c_{3} \operatorname {MathieuCPrime}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {\lambda x}{2}\right )+\operatorname {MathieuSPrime}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right )}{2 \alpha \left (c_{3} \operatorname {MathieuC}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {\lambda x}{2}\right )+\operatorname {MathieuS}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \alpha \gamma }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {\lambda \left (c_{1} \operatorname {MathieuSPrime}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \gamma \alpha }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {x \lambda }{2}\right )+\operatorname {MathieuCPrime}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \gamma \alpha }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {x \lambda }{2}\right )\right )}{2 \alpha \left (c_{1} \operatorname {MathieuS}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \gamma \alpha }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {x \lambda }{2}\right )+\operatorname {MathieuC}\left (\frac {4 \alpha \beta }{\lambda ^{2}}, -\frac {2 \gamma \alpha }{\lambda ^{2}}, -\frac {\pi }{4}+\frac {x \lambda }{2}\right )\right )} \]

Problem 10499


ODE	\[ \boxed {y^{\prime }-y^{2}=-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2}} \]

program solution	\[ y = \frac {-\left (\cos \left (\lambda x \right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) a +\frac {\lambda \cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )-\cos \left (\lambda x \right ) \left (\frac {\lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )}{2}+\operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{3} a +\frac {c_{3} \lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}\right )}{c_{3} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (\lambda x \right )}{2}+\frac {1}{2}\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (-2 a c_{1} \cos \left (x \lambda \right ) \sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )-c_{1} \lambda \cos \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )-2 \left (a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )+\frac {\lambda \left (\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right ) c_{1} \sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )+\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )\right )}{2}\right ) \cos \left (x \lambda \right )}{2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right ) c_{1} +2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )} \]

Problem 10500


ODE	\[ \boxed {y^{\prime }-y^{2}=\lambda ^{2}+c \sin \left (\lambda x +a \right )^{n} \sin \left (\lambda x +b \right )^{-n -4}} \]

program solution

Maple solution	\[ \text {No solution found} \]

2.17.105 Problems 10401 to 10500