Problems 10501 to 10600

Problem 10501


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

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

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

Problem 10502


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

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

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

Problem 10503


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

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

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

Problem 10504


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

program solution	\[ y = -\frac {\sqrt {2}\, \left (\sqrt {\sin \left (\lambda x \right )-1}\, \left (a \cos \left (\lambda x \right )^{2}+2 \left (a +\frac {\lambda }{2}\right ) \left (\sin \left (\lambda x \right )-1\right )\right ) \sqrt {2}\, \left (a \cos \left (\lambda x \right )+\tan \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \lambda \right ) \left (\int _{}^{\sin \left (\lambda x \right )}\frac {\left (\left (\textit {\_a} -1\right ) a -\lambda \right ) {\mathrm e}^{\frac {\textit {\_a} a}{\lambda }}}{\left (\textit {\_a} -1\right )^{\frac {3}{2}} \sqrt {\textit {\_a} +1}}d \textit {\_a} \right )+\sqrt {\sin \left (\lambda x \right )-1}\, \left (a \cos \left (\lambda x \right )^{2}+2 \left (a +\frac {\lambda }{2}\right ) \left (\sin \left (\lambda x \right )-1\right )\right ) \sqrt {2}\, \left (a \cos \left (\lambda x \right )+\tan \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \lambda \right ) c_{3} -2 \csc \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \left (\cos \left (\lambda x \right )^{2} a^{2}+2 a \left (a +\lambda \right ) \sin \left (\lambda x \right )-2 a^{2}-2 \lambda a -\lambda ^{2}\right ) \lambda \,{\mathrm e}^{\frac {a \sin \left (\lambda x \right )}{\lambda }} \cos \left (\lambda x \right )\right ) \sec \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )^{2}}{4 \sqrt {\sin \left (\lambda x \right )-1}\, \left (a \sin \left (\lambda x \right )-a -\lambda \right )^{2} \left (c_{3} +\int _{}^{\sin \left (\lambda x \right )}\frac {\left (\left (\textit {\_a} -1\right ) a -\lambda \right ) {\mathrm e}^{\frac {\textit {\_a} a}{\lambda }}}{\left (\textit {\_a} -1\right )^{\frac {3}{2}} \sqrt {\textit {\_a} +1}}d \textit {\_a} \right )} \] Warning, solution could not be veriﬁed

Maple solution	\[ y \left (x \right ) = -\frac {\left (\left (\left (\int _{}^{\sin \left (x \lambda \right )}\frac {\left (a \left (\textit {\_a} -1\right )-\lambda \right ) {\mathrm e}^{\frac {a \textit {\_a}}{\lambda }}}{\left (\textit {\_a} -1\right )^{\frac {3}{2}} \sqrt {\textit {\_a} +1}}d \textit {\_a} \right ) c_{1} +1\right ) \sqrt {-\cos \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )^{2}}\, \left (a \cos \left (x \lambda \right )+\tan \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right ) \lambda \right ) \operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )\right )+\frac {\sec \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )^{2} \csc \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right ) \cos \left (x \lambda \right ) {\mathrm e}^{\frac {a \sin \left (x \lambda \right )}{\lambda }} c_{1} \lambda \left (-\lambda -a +a \sin \left (x \lambda \right )\right )}{2}\right ) \operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )\right )}{\sqrt {-\cos \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )^{2}}\, \left (\left (\int _{}^{\sin \left (x \lambda \right )}\frac {\left (a \left (\textit {\_a} -1\right )-\lambda \right ) {\mathrm e}^{\frac {a \textit {\_a}}{\lambda }}}{\left (\textit {\_a} -1\right )^{\frac {3}{2}} \sqrt {\textit {\_a} +1}}d \textit {\_a} \right ) c_{1} +1\right ) \left (-\lambda -a +a \sin \left (x \lambda \right )\right )} \]

Problem 10505


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

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

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

Problem 10506


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

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

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

Problem 10507


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

program solution	\[ y = b \,x^{n}+c \] Veriﬁed OK.

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

Problem 10508


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

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

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

Problem 10509


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

program solution

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

Problem 10510


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

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

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

Problem 10511


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

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

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

Problem 10512


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

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

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

Problem 10513


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

program solution

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

Problem 10514


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

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

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

Problem 10515


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

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

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

Problem 10516


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

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

Maple solution	\[ y \left (x \right ) = -\frac {4 \csc \left (x \lambda \right ) \left (-\frac {\sqrt {\pi }\, \sin \left (x \lambda \right )^{2} \left (c_{1} -\frac {1}{2}\right ) \operatorname {erf}\left (\sqrt {-\sin \left (x \lambda \right )^{2}}\right )}{2}+\left (c_{1} -\frac {1}{2}\right ) {\mathrm e}^{\sin \left (x \lambda \right )^{2}} \sqrt {-\sin \left (x \lambda \right )^{2}}+\frac {\sin \left (x \lambda \right )^{2} \sqrt {\pi }\, c_{1}}{2}\right )}{\sqrt {\pi }\, \left (\operatorname {erf}\left (\sqrt {-\sin \left (x \lambda \right )^{2}}\right ) \left (2 c_{1} -1\right )-2 c_{1} \right )} \]

Problem 10517


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

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

Maple solution	\[ y \left (x \right ) = \frac {-\cot \left (\frac {x \lambda }{2}\right ) \lambda \left (\int {\mathrm e}^{\frac {a \cos \left (x \lambda \right )}{\lambda }} \operatorname {csgn}\left (\sin \left (\frac {x \lambda }{2}\right )\right ) \left (\csc \left (\frac {x \lambda }{2}\right )^{2} \lambda +2 a \right )d x \right ) c_{1} -2 \csc \left (\frac {x \lambda }{2}\right )^{2} \operatorname {csgn}\left (\sin \left (\frac {x \lambda }{2}\right )\right ) {\mathrm e}^{\frac {a \cos \left (x \lambda \right )}{\lambda }} c_{1} \lambda +2 i \cot \left (\frac {x \lambda }{2}\right )}{\lambda \left (\int {\mathrm e}^{\frac {a \cos \left (x \lambda \right )}{\lambda }} \operatorname {csgn}\left (\sin \left (\frac {x \lambda }{2}\right )\right ) \left (\csc \left (\frac {x \lambda }{2}\right )^{2} \lambda +2 a \right )d x \right ) c_{1} -2 i} \]

Problem 10518


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

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

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

Problem 10519


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

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

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

Problem 10520


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

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

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

Problem 10521


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

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

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

Problem 10522


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

program solution

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

Problem 10523


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

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

Maple solution	\[ y \left (x \right ) = \frac {\left (2 \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right ) \sqrt {a^{2}-b^{2}}\, a b \cos \left (\frac {x \lambda }{2}\right ) \sin \left (\frac {x \lambda }{2}\right )-2 \sqrt {a^{2}-b^{2}}\, c_{1} a \cos \left (\frac {x \lambda }{2}\right ) \sin \left (\frac {x \lambda }{2}\right )+\left (a \cos \left (\frac {x \lambda }{2}\right )^{2}-\frac {a}{2}-\frac {b}{2}\right ) \left (a +b \right ) \left (a -b \right )\right ) \lambda }{\sqrt {a^{2}-b^{2}}\, \left (2 \left (a \cos \left (\frac {x \lambda }{2}\right )^{2}-\frac {a}{2}+\frac {b}{2}\right ) b \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )-\sqrt {a^{2}-b^{2}}\, a \cos \left (\frac {x \lambda }{2}\right ) \sin \left (\frac {x \lambda }{2}\right )-2 c_{1} \left (a \cos \left (\frac {x \lambda }{2}\right )^{2}-\frac {a}{2}+\frac {b}{2}\right )\right )} \]

Problem 10524


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

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

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

Problem 10525


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

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

Maple solution	\[ y \left (x \right ) = -\frac {\left (-2 \operatorname {LegendreP}\left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (x \lambda \right )\right ) \lambda -2 \operatorname {LegendreQ}\left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (x \lambda \right )\right ) c_{1} \lambda +\sin \left (x \lambda \right ) \left (\operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (x \lambda \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (x \lambda \right )\right )\right ) \left (a +\lambda \right )\right ) \sec \left (x \lambda \right )}{\operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (x \lambda \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (x \lambda \right )\right )} \]

Problem 10526


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

program solution	\[ y = -\frac {\left (\left (\sqrt {4 a c +b^{2}}+b \right ) \left (c_{3} \operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )+\operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )\right ) \sin \left (x \right )-\left (\sqrt {4 a c +b^{2}}-b +2\right ) \left (c_{3} \operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}+\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )+\operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}+\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )\right )\right ) \sec \left (x \right )}{2 \left (c_{3} \operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )+\operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )\right ) a} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\sec \left (x \right ) \left (-\left (\operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )\right ) \left (b +\sqrt {4 a c +b^{2}}\right ) \sin \left (x \right )+\left (\sqrt {4 a c +b^{2}}-b +2\right ) \left (\operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}+\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}+\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )\right )\right )}{2 \left (\operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )\right ) a} \]

Problem 10527


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

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

Maple solution	\[ y \left (x \right ) = \frac {2 c_{1} a b -2 i \tan \left (x \right ) a^{\frac {3}{2}} b^{\frac {3}{2}} c_{1} +i \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x}-\tan \left (x \right ) {\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x} a b}{a \left (2 i c_{1} \sqrt {a}\, \sqrt {b}+{\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x}\right )} \]

Problem 10528


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

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

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

Problem 10529


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

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

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

Problem 10530


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

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

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

Problem 10531


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

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

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

Problem 10532


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

program solution

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

Problem 10533


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

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

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

Problem 10534


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

program solution

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

Problem 10535


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

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

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

Problem 10536


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

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

Maple solution	\[ y \left (x \right ) = \frac {\csc \left (x \lambda \right ) \left (-2 \operatorname {LegendreP}\left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (x \lambda \right )\right ) \lambda -2 \operatorname {LegendreQ}\left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (x \lambda \right )\right ) c_{1} \lambda +\cos \left (x \lambda \right ) \left (\operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (x \lambda \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (x \lambda \right )\right )\right ) \left (a +\lambda \right )\right )}{\operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (x \lambda \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (x \lambda \right )\right )} \]

Problem 10537


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

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

Maple solution	\[ y \left (x \right ) = \frac {\left (\cos \left (a x \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )+c_{1} \cos \left (a x \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )+\left (\operatorname {LegendreQ}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )\right ) \left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}+\left (b -1\right ) a \right )\right ) \csc \left (a x \right )}{\operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )} \]

Problem 10538


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

program solution	\[ y = \frac {-2 i b c_{3} -\beta \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}-\beta \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right ) b}{\beta \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right )+2 i c_{3}} \] Veriﬁed OK.

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

Problem 10539


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

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

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

Problem 10540


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

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

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

Problem 10541


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

program solution

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

Problem 10542


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

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

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

Problem 10543


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

program solution

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

Problem 10544


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

program solution	\[ y = -\frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{\cos \left (\lambda x \right )^{-n -4} \sin \left (\lambda x \right )^{n} \textit {\_Y} \left (x \right ) c +\textit {\_Y} \left (x \right ) \lambda ^{2}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\cos \left (\lambda x \right )^{-n -4} \sin \left (\lambda x \right )^{n} \textit {\_Y} \left (x \right ) c +\textit {\_Y} \left (x \right ) \lambda ^{2}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

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

Problem 10545


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

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

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

Problem 10546


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

program solution	\[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{-\sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n -1} a \lambda \textit {\_Y} \left (x \right )-\cos \left (\lambda x \right )^{n} \textit {\_Y}^{\prime }\left (x \right ) a -\textit {\_Y}^{\prime }\left (x \right ) \lambda \cot \left (\lambda x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \csc \left (\lambda x \right )}{\lambda \operatorname {DESol}\left (\left \{-\sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n -1} a \lambda \textit {\_Y} \left (x \right )-\cos \left (\lambda x \right )^{n} \textit {\_Y}^{\prime }\left (x \right ) a -\textit {\_Y}^{\prime }\left (x \right ) \lambda \cot \left (\lambda x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Veriﬁed OK.

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

Problem 10547


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

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

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

Problem 10548


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

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

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

Problem 10549


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

program solution	\[ y = -\frac {\csc \left (x \right ) \sin \left (x \right )^{-2 n} \sin \left (2 x \right )^{n +1} \left (\left (n +\sqrt {n^{2}-4^{-n} b a}\right ) \cot \left (x \right )^{-\frac {\sqrt {n^{2}-4^{-n} b a}}{2}}-\cot \left (x \right )^{\frac {\sqrt {n^{2}-4^{-n} b a}}{2}} c_{3} \left (-n +\sqrt {n^{2}-4^{-n} b a}\right )\right ) \sec \left (x \right )}{2 \left (c_{3} \cot \left (x \right )^{\frac {\sqrt {n^{2}-4^{-n} b a}}{2}}+\cot \left (x \right )^{-\frac {\sqrt {n^{2}-4^{-n} b a}}{2}}\right ) a} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {\csc \left (x \right ) \sin \left (2 x \right )^{n} \left (\sin \left (x \right )^{\frac {\sqrt {n^{2}-4^{-n} a b}}{2}} \left (n +\sqrt {n^{2}-4^{-n} a b}\right ) \cos \left (x \right )^{-\frac {\sqrt {n^{2}-4^{-n} a b}}{2}}-\cos \left (x \right )^{\frac {\sqrt {n^{2}-4^{-n} a b}}{2}} \sin \left (x \right )^{-\frac {\sqrt {n^{2}-4^{-n} a b}}{2}} c_{1} \left (-n +\sqrt {n^{2}-4^{-n} a b}\right )\right ) \sin \left (x \right )^{-2 n +1}}{a \left (\cos \left (x \right )^{\frac {\sqrt {n^{2}-4^{-n} a b}}{2}} \sin \left (x \right )^{-\frac {\sqrt {n^{2}-4^{-n} a b}}{2}} c_{1} +\cos \left (x \right )^{-\frac {\sqrt {n^{2}-4^{-n} a b}}{2}} \sin \left (x \right )^{\frac {\sqrt {n^{2}-4^{-n} a b}}{2}}\right )} \]

Problem 10550


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

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

Maple solution	\[ y \left (x \right ) = \frac {-\cot \left (x \right ) \sin \left (x \right )^{2 a} a +c_{1} \cos \left (x \right ) \left (a -1\right )}{c_{1} \sin \left (x \right )+\sin \left (x \right )^{2 a}} \]

Problem 10551


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

program solution	\[ y = -\frac {\left (\sin \left (x \right )^{2} \left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right )-3 \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \left (\cos \left (x \right ) \sin \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m -2}}-\cos \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m}}\, c_{3} \right ) \cos \left (x \right )^{-m +1} b}{3 c_{3} \sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+3 \cos \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (-\frac {\sin \left (x \right )^{2} \left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) b \cos \left (x \right )^{-m +1} \left (\sin \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right ) c_{1} -\sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )\right )}{c_{1} \cos \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+\sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )} \]

Problem 10552


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

program solution	\[ y = \frac {\left (-\sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+c_{3} \cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \left (\csc \left (x \right )^{2}\right )^{-\frac {m}{2}} \left (-\frac {\cot \left (x \right )^{2} \left (2+m \right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, 2+\frac {m}{2}\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right ) b}{\left (c_{3} \sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+\cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \left (-c_{1} \sin \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+\cos \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} b \csc \left (x \right )^{6} \left (-\frac {\operatorname {hypergeom}\left (\left [\frac {3}{2}, 2+\frac {m}{2}\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right ) \cos \left (x \right )^{2} \left (m +2\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) \sin \left (x \right )^{2}\right )}{c_{1} \cos \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+\sin \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )} \]

Problem 10553


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

program solution

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

Problem 10554


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

program solution	\[ y = \frac {4 \lambda \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{2} \left (a +b -\lambda \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -a -b}{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+\left (\left (6 \lambda ^{2}+\left (-7 a -3 b \right ) \lambda +2 b a \right ) \cos \left (\lambda x \right )^{2}-2 \sin \left (\lambda x \right )^{2} a^{2}+5 a \lambda -3 \lambda ^{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+2 \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }} c_{3} \left (\tan \left (\lambda x \right ) a -b \cot \left (\lambda x \right )\right ) \left (-\frac {3 \lambda }{2}+a \right ) \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }}}{\left (-3 \lambda +2 a \right ) \left (c_{3} \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }} \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }}+\cos \left (\lambda x \right ) \sin \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )\right )} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {4 c_{1} \lambda \cos \left (x \lambda \right )^{2} \sin \left (x \lambda \right )^{2} \left (b -\lambda +a \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -b -a}{\lambda }\right ], \left [-\frac {2 a -5 \lambda }{2 \lambda }\right ], \cos \left (x \lambda \right )^{2}\right )-2 c_{1} \left (\left (-3 \lambda ^{2}+\left (\frac {7 a}{2}+\frac {3 b}{2}\right ) \lambda -a b \right ) \cos \left (x \lambda \right )^{2}+a^{2} \sin \left (x \lambda \right )^{2}-\frac {5 \left (a -\frac {3 \lambda }{5}\right ) \lambda }{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {2 a -3 \lambda }{2 \lambda }\right ], \cos \left (x \lambda \right )^{2}\right )+2 \left (a -\frac {3 \lambda }{2}\right ) \sin \left (x \lambda \right )^{\frac {2 b}{\lambda }} \left (a \tan \left (x \lambda \right )-b \cot \left (x \lambda \right )\right ) \cos \left (x \lambda \right )^{\frac {2 a}{\lambda }}}{\left (2 a -3 \lambda \right ) \left (c_{1} \cos \left (x \lambda \right ) \sin \left (x \lambda \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {2 a -3 \lambda }{2 \lambda }\right ], \cos \left (x \lambda \right )^{2}\right )+\cos \left (x \lambda \right )^{\frac {2 a}{\lambda }} \sin \left (x \lambda \right )^{\frac {2 b}{\lambda }}\right )} \]

Problem 10555


ODE	\[ \boxed {y^{\prime }-y^{2}=-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n}} \]

program solution	\[ y = \frac {\lambda \left (-2 \Gamma \left (\frac {n +3}{n +2}\right )^{2} \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +1}{2 n +4}} \cos \left (\lambda x \right ) \left (n +2\right )^{2} \operatorname {BesselI}\left (\frac {n +3}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right )+2 \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +3}{2 n +4}} \pi c_{3} \cot \left (\lambda x \right ) \left (n +2\right ) \operatorname {BesselI}\left (\frac {n +1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right )+\tan \left (\lambda x \right ) c_{3} \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \pi \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}-\Gamma \left (\frac {n +3}{n +2}\right )^{2} \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \left (\cos \left (\lambda x \right )+\sec \left (\lambda x \right )\right ) \left (n +2\right )\right )}{-2 c_{3} \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \pi \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}+2 \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \Gamma \left (\frac {n +3}{n +2}\right )^{2} \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \sin \left (\lambda x \right ) \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (n +2\right )} \] Veriﬁed OK.

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

Problem 10556


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

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

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

Problem 10557


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

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

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

Problem 10558


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

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

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

Problem 10559


ODE	\[ \boxed {y^{\prime }+\left (k +1\right ) x^{k} y^{2}-\lambda \arcsin \left (x \right )^{n} \left (x^{k +1} y-1\right )=0} \]

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

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

Problem 10560


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

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

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

Problem 10561


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

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

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

Problem 10562


ODE	\[ \boxed {y^{\prime }-\lambda \arcsin \left (x \right )^{n} y^{2}=\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arcsin \left (x \right )^{n}} \]

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

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

Problem 10563


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

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

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

Problem 10564


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

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

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

Problem 10565


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

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

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

Problem 10566


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

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

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

Problem 10567


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

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

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

Problem 10568


ODE	\[ \boxed {y^{\prime }+\left (k +1\right ) x^{k} y^{2}-\lambda \arccos \left (x \right )^{n} \left (x^{k +1} y-1\right )=0} \]

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

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

Problem 10569


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

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

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

Problem 10570


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

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

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

Problem 10571


ODE	\[ \boxed {y^{\prime }-\lambda \arccos \left (x \right )^{n} y^{2}=\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n}} \]

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

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

Problem 10572


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

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

Maple solution	\[ y \left (x \right ) = \frac {\lambda \left (a \,x^{m}+b \right ) \left (\left (n +2\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {3}{2}}\right ) \sqrt {-x^{2}+1}-\left (\lambda \arccos \left (x \right ) \left (a \,x^{1+m}+b x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\sqrt {\arccos \left (x \right )}\, \left (x^{m} c_{1} a +c_{1} b +1\right )\right ) \left (n +2\right )}{\lambda \left (\left (n +2\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {3}{2}}\right ) \sqrt {-x^{2}+1}+\left (n +2\right ) \left (-x \lambda \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+c_{1} \sqrt {\arccos \left (x \right )}\right )} \]

Problem 10573


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

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

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

Problem 10574


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

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

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

Problem 10575


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

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

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

Problem 10576


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

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

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

Problem 10577


ODE	\[ \boxed {y^{\prime }+\left (k +1\right ) x^{k} y^{2}-\lambda \arctan \left (x \right )^{n} \left (x^{k +1} y-1\right )=0} \]

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

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

Problem 10578


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

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

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

Problem 10579


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

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

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

Problem 10580


ODE	\[ \boxed {y^{\prime }-\lambda \arctan \left (x \right )^{n} y^{2}=\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arctan \left (x \right )^{n}} \]

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

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

Problem 10581


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

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

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

Problem 10582


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

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

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

Problem 10583


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

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

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

Problem 10584


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

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

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

Problem 10585


ODE	\[ \boxed {y^{\prime }-y^{2}-\lambda x \operatorname {arccot}\left (x \right )^{n} y=\operatorname {arccot}\left (x \right )^{n} \lambda } \]

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

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

Problem 10586


ODE	\[ \boxed {y^{\prime }+\left (k +1\right ) x^{k} y^{2}-\lambda \operatorname {arccot}\left (x \right )^{n} \left (x^{k +1} y-1\right )=0} \]

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

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

Problem 10587


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

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

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

Problem 10588


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

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

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

Problem 10589


ODE	\[ \boxed {y^{\prime }-\lambda \operatorname {arccot}\left (x \right )^{n} y^{2}=\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \operatorname {arccot}\left (x \right )^{n}} \]

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

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

Problem 10590


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

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

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

Problem 10591


ODE	\[ \boxed {x y^{\prime }-\lambda \operatorname {arccot}\left (x \right )^{n} y^{2}-y k=\lambda \,b^{2} x^{2 k} \operatorname {arccot}\left (x \right )^{n}} \]

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

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

Problem 10592


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

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

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

Problem 10593


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

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

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

Problem 10594


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

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

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

Problem 10595


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

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

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

Problem 10596


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

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

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

Problem 10597


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

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

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

Problem 10598


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

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

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

Problem 10599


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

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

Maple solution	\[ y \left (x \right ) = -\tan \left (-\sqrt {a}\, \left (\int f \left (x \right ) x^{n -1}d x \right )+c_{1} \right ) \sqrt {a}\, x^{n} \]

Problem 10600


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

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

Maple solution	\[ y \left (x \right ) = -\frac {\left (a^{2}+\tanh \left (\frac {\sqrt {a^{2} \left (a^{2}-4 b \right )}\, \left (a \left (\int f \left (x \right ) x^{n -1}d x \right )+c_{1} \right )}{2 a^{2}}\right ) \sqrt {a^{2} \left (a^{2}-4 b \right )}\right ) x^{-n}}{2 a} \]

2.17.106 Problems 10501 to 10600