2.17.45 Problems 4401 to 4500

Problem 4401

ODE	\[ \boxed {x y^{\prime }-a y+y^{2}=x^{-\frac {2 a}{3}}} \]
program solution	\[ y = \frac {\left (\left (-2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a -x^{-\frac {a}{3}}\right )\right ) c_{3} {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+\left (2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a +x^{-\frac {a}{3}}\right )}{c_{3} \left (-3 \sqrt {x^{-\frac {2 a}{3}}}+a \right ) {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+3 \sqrt {x^{-\frac {2 a}{3}}}+a} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (-2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a -x^{-\frac {a}{3}}\right )\right ) {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+\left (\left (2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a +x^{-\frac {a}{3}}\right )\right ) c_{1}}{\left (-3 \sqrt {x^{-\frac {2 a}{3}}}+a \right ) {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+c_{1} \left (3 \sqrt {x^{-\frac {2 a}{3}}}+a \right )} \]

Problem 4402

ODE	\[ \boxed {u^{\prime }+u^{2}=\frac {c}{x^{\frac {4}{3}}}} \]
program solution	\[ u = -\frac {3 c \left (-{\mathrm e}^{6 x^{\frac {1}{3}} \sqrt {c}}+c_{3} \right )}{x^{\frac {1}{3}} \left (3 x^{\frac {1}{3}} \sqrt {c}\, {\mathrm e}^{6 x^{\frac {1}{3}} \sqrt {c}}+3 \sqrt {c}\, x^{\frac {1}{3}} c_{3} -{\mathrm e}^{6 x^{\frac {1}{3}} \sqrt {c}}+c_{3} \right )} \] Verified OK.
Maple solution	\[ u \left (x \right ) = -\frac {3 c}{x^{\frac {1}{3}} \left (3 x^{\frac {1}{3}} \tan \left (3 \sqrt {-c}\, \left (x^{\frac {1}{3}}-c_{1} \right )\right ) \sqrt {-c}+1\right )} \]

Problem 4403

ODE	\[ \boxed {u^{\prime }+b u^{2}=\frac {c}{x^{4}}} \]
program solution	\[ u = \frac {\left (-c_{3} \sqrt {b c}+x \right ) \cosh \left (\frac {\sqrt {b c}}{x}\right )+\sinh \left (\frac {\sqrt {b c}}{x}\right ) \left (c_{3} x -\sqrt {b c}\right )}{x^{2} b \left (c_{3} \sinh \left (\frac {\sqrt {b c}}{x}\right )+\cosh \left (\frac {\sqrt {b c}}{x}\right )\right )} \] Verified OK.
Maple solution	\[ u \left (x \right ) = \frac {-\sqrt {-b c}\, \tan \left (\frac {\sqrt {-b c}\, \left (c_{1} x -1\right )}{x}\right )+x}{b \,x^{2}} \]

Problem 4404

ODE	\[ \boxed {u^{\prime }-u^{2}=\frac {2}{x^{\frac {8}{3}}}} \]
program solution	\[ u = \frac {\left (3 i x^{\frac {2}{3}} \sqrt {2}+54 i \sqrt {2}+5 x +36 x^{\frac {1}{3}}+\frac {x^{\frac {5}{3}}}{6}\right ) {\mathrm e}^{-\frac {3 i \sqrt {2}}{x^{\frac {1}{3}}}}+\sqrt {x^{\frac {1}{3}} \sqrt {2}+6 i}\, {\mathrm e}^{\frac {3 i \sqrt {2}}{x^{\frac {1}{3}}}} \left (i x^{\frac {2}{3}} \sqrt {2}-3 i \sqrt {2}-\frac {x}{6}+4 x^{\frac {1}{3}}\right ) c_{3} \sqrt {x^{\frac {2}{3}}+18}\, \sqrt {x^{\frac {1}{3}} \sqrt {2}-6 i}}{3 \sqrt {x^{\frac {2}{3}}+18}\, x^{\frac {4}{3}} \sqrt {x^{\frac {1}{3}} \sqrt {2}+6 i}\, \left (-\frac {x^{\frac {1}{3}} \sqrt {2}}{6}+i\right ) \left (\frac {{\mathrm e}^{-\frac {3 i \sqrt {2}}{x^{\frac {1}{3}}}} \sqrt {x^{\frac {2}{3}}+18}\, \sqrt {x^{\frac {1}{3}} \sqrt {2}+6 i}}{6}+\left (-\frac {x^{\frac {1}{3}} \sqrt {2}}{6}+i\right ) c_{3} \sqrt {x^{\frac {1}{3}} \sqrt {2}-6 i}\, {\mathrm e}^{\frac {3 i \sqrt {2}}{x^{\frac {1}{3}}}}\right )} \] Verified OK.
Maple solution	\[ u \left (x \right ) = -\frac {3 \left (\tan \left (3 \sqrt {2}\, \left (\left (\frac {1}{x}\right )^{\frac {1}{3}}-c_{1} \right )\right ) \sqrt {2}\, x \left (\frac {1}{x}\right )^{\frac {2}{3}}+\frac {x \left (\frac {1}{x}\right )^{\frac {1}{3}}}{3}-2\right )}{\left (\frac {1}{x}\right )^{\frac {1}{3}} x^{2} \left (3 \left (\frac {1}{x}\right )^{\frac {1}{3}} \sqrt {2}\, \tan \left (3 \sqrt {2}\, \left (\left (\frac {1}{x}\right )^{\frac {1}{3}}-c_{1} \right )\right )+1\right )} \]

Problem 4405

ODE	\[ \boxed {\frac {\sqrt {f \,x^{4}+c \,x^{3}+c \,x^{2}+b x +a}\, y^{\prime }}{\sqrt {a +b y+y^{2} c +c y^{3}+f y^{4}}}=-1} \]
program solution	\[ \text {Expression too large to display} \] Warning, solution could not be verified
Maple solution	\[ \int \frac {1}{\sqrt {f \,x^{4}+x^{3} c +x^{2} c +x b +a}}d x +\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{4} f +\textit {\_a}^{3} c +\textit {\_a}^{2} c +\textit {\_a} b +a}}d \textit {\_a} +c_{1} = 0 \]

Problem 4406

ODE	\[ \boxed {{y^{\prime }}^{2}-5 y^{\prime }=-6} \]
program solution	\[ y = 3 x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 3 x +c_{1} \\ y \left (x \right ) &= 2 x +c_{1} \\ \end{align*}

Problem 4407

ODE	\[ \boxed {{y^{\prime }}^{2}=\frac {a^{2}}{x^{2}}} \]
program solution	\[ y = -a \ln \left (x \right )+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= a \ln \left (x \right )+c_{1} \\ y \left (x \right ) &= -a \ln \left (x \right )+c_{1} \\ \end{align*}

Problem 4408

ODE	\[ \boxed {{y^{\prime }}^{2}=\frac {1-x}{x}} \]
program solution	\[ y = \sqrt {-x^{2}+x}+\frac {\arcsin \left (2 x -1\right )}{2}+c_{1} \] Verified OK. \[ y = -\sqrt {-x^{2}+x}-\frac {\arcsin \left (2 x -1\right )}{2}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {-x \left (x -1\right )}+\frac {\arcsin \left (2 x -1\right )}{2}+c_{1} \\ y \left (x \right ) &= -\sqrt {-x \left (x -1\right )}-\frac {\arcsin \left (2 x -1\right )}{2}+c_{1} \\ \end{align*}

Problem 4409

ODE	\[ \boxed {{y^{\prime }}^{2}=-\frac {2 x y^{\prime }}{y}+1} \]
program solution	\[ y = 0 \] Verified OK. \[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ x = -\frac {2 c_{3} x}{-x +\sqrt {x^{2}+y^{2}}} \] Verified OK. \[ x = \frac {2 c_{3} x}{\sqrt {x^{2}+y^{2}}+x} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= -\frac {2 \sqrt {c_{1} x +1}}{c_{1}} \\ y \left (x \right ) &= \frac {2 \sqrt {c_{1} x +1}}{c_{1}} \\ \end{align*}

Problem 4410

ODE	\[ \boxed {y-a y^{\prime }-b {y^{\prime }}^{2}=0} \]
program solution	\[ \sqrt {a^{2}+4 b y}+a \ln \left (a -\sqrt {a^{2}+4 b y}\right ) = x +c_{1} \] Verified OK. \[ -\sqrt {a^{2}+4 b y}+a \ln \left (a +\sqrt {a^{2}+4 b y}\right ) = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= {\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a \sqrt {\frac {1}{b}}}\right )-a +x -c_{1}}{a}} \left (a \sqrt {\frac {1}{b}}+{\mathrm e}^{\frac {-a \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a \sqrt {\frac {1}{b}}}\right )-a +x -c_{1}}{a}}\right ) \\ y \left (x \right ) &= \frac {a^{2} \left (\operatorname {LambertW}\left (-\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (-\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a}\right )}{4 b} \\ y \left (x \right ) &= \frac {a^{2} \left (\operatorname {LambertW}\left (\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a}\right )+2\right ) \operatorname {LambertW}\left (\frac {2 \sqrt {b}\, {\mathrm e}^{\frac {-c_{1} -a +x}{a}}}{a}\right )}{4 b} \\ \end{align*}

Problem 4411

ODE	\[ \boxed {-a y^{\prime }-b {y^{\prime }}^{2}=-x} \]
program solution	\[ y = \frac {\frac {\left (a^{2}+4 b x \right )^{\frac {3}{2}}}{6 b}-a x}{2 b}+c_{1} \] Verified OK. \[ y = -\frac {a x +\frac {\left (a^{2}+4 b x \right )^{\frac {3}{2}}}{6 b}}{2 b}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (a^{2}+4 x b \right )^{\frac {3}{2}}+12 c_{1} b^{2}-6 a x b}{12 b^{2}} \\ y \left (x \right ) &= \frac {12 c_{1} b^{2}-a^{2} \sqrt {a^{2}+4 x b}-6 a x b -4 b x \sqrt {a^{2}+4 x b}}{12 b^{2}} \\ \end{align*}

Problem 4412

ODE	\[ \boxed {y-\sqrt {1+{y^{\prime }}^{2}}-a y^{\prime }=0} \]
program solution	\[ \int _{}^{y}\frac {a^{2}-1}{a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}\frac {a^{2}-1}{a \textit {\_a} -\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} -\left (\int _{}^{y \left (x \right )}\frac {1}{a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right ) a^{2}+\int _{}^{y \left (x \right )}\frac {1}{a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} -c_{1} +x &= 0 \\ \left (\int _{}^{y \left (x \right )}\frac {1}{-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right ) a^{2}-\left (\int _{}^{y \left (x \right )}\frac {1}{-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right )-c_{1} +x &= 0 \\ \end{align*}

Problem 4413

ODE	\[ \boxed {-\sqrt {1+{y^{\prime }}^{2}}-a y^{\prime }=-x} \]
program solution	\[ y = \frac {\frac {x \sqrt {a^{2}+x^{2}-1}}{2}+\frac {\left (4 a^{2}-4\right ) \ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )}{8}+\frac {a \,x^{2}}{2}}{a^{2}-1}+c_{1} \] Verified OK. \[ y = -\frac {\frac {x \sqrt {a^{2}+x^{2}-1}}{2}+\frac {\left (4 a^{2}-4\right ) \ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )}{8}-\frac {a \,x^{2}}{2}}{a^{2}-1}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {a \,x^{2}+x \sqrt {a^{2}+x^{2}-1}+\left (\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )+2 c_{1} \right ) \left (1+a \right ) \left (a -1\right )}{2 a^{2}-2} \\ y \left (x \right ) &= \frac {a \,x^{2}-x \sqrt {a^{2}+x^{2}-1}-\left (1+a \right ) \left (a -1\right ) \left (\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )-2 c_{1} \right )}{2 a^{2}-2} \\ \end{align*}

Problem 4414

ODE	\[ \boxed {y^{\prime }-\frac {\sqrt {1+{y^{\prime }}^{2}}}{x}=0} \]
program solution	\[ y = \ln \left (x +\sqrt {x^{2}-1}\right )+c_{1} \] Verified OK. \[ y = \ln \left (x -\sqrt {x^{2}-1}\right )+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \ln \left (x +\sqrt {x^{2}-1}\right )+c_{1} \\ y \left (x \right ) &= -\ln \left (x +\sqrt {x^{2}-1}\right )+c_{1} \\ \end{align*}

Problem 4415

ODE	\[ \boxed {x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}=a^{2}} \]
program solution	\[ y = -\frac {\sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{4}}}\, \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \] Verified OK. \[ y = \frac {\sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{4}}}\, \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{2} \] Verified OK. \[ y = -\frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{3} \] Verified OK. \[ y = \frac {\sqrt {2}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{4} \] Verified OK. \[ y = \frac {\sqrt {-\frac {2 \left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{5} \] Verified OK. \[ y = -\frac {\sqrt {-\frac {2 \left (a^{2} x \right )^{\frac {4}{3}} \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{a^{4}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{6} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-\sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}\, a^{2}+c_{1} \left (a^{2} x \right )^{\frac {2}{3}}+\sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}\, \left (a^{2} x \right )^{\frac {2}{3}}}{\left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right ) \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}+c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{\left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {\left (2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}-\sqrt {3}\, a^{2}\right ) x \left (a^{2} x \right )^{\frac {1}{3}}}{a^{2}}}\, \left (2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}+i \sqrt {3}\, a^{2}\right )}{4 \sqrt {\left (i \left (a^{2} x \right )^{\frac {1}{3}}+2 i x -\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}\right ) x}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {\left (2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}-\sqrt {3}\, a^{2}\right ) x \left (a^{2} x \right )^{\frac {1}{3}}}{a^{2}}}\, \left (2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}+i \sqrt {3}\, a^{2}\right )}{4 \sqrt {\left (i \left (a^{2} x \right )^{\frac {1}{3}}+2 i x -\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}\right ) x}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \left (x \right ) &= \frac {\left (-2 \left (a^{2} x \right )^{\frac {2}{3}} \sqrt {2}+\left (i \sqrt {6}-\sqrt {2}\right ) a^{2}\right ) \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}\right ) x \left (a^{2} x \right )^{\frac {1}{3}}}{a^{2}}}+4 c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{4 \left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (-2 \left (a^{2} x \right )^{\frac {2}{3}} \sqrt {2}+\left (i \sqrt {6}-\sqrt {2}\right ) a^{2}\right ) \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}\right ) x \left (a^{2} x \right )^{\frac {1}{3}}}{a^{2}}}-4 c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{4 \left (a^{2} x \right )^{\frac {2}{3}}} \\ \end{align*}

Problem 4416

ODE	\[ \boxed {{y^{\prime }}^{2}=-1+\frac {\left (a +x \right )^{2}}{2 a x +x^{2}}} \]
program solution	\[ y = a \ln \left (a +x +\sqrt {2 a x +x^{2}}\right )+c_{1} \] Verified OK. \[ y = -2 a \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (2 a +x \right )}}{x}\right )+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= a \ln \left (x +a +\sqrt {x \left (2 a +x \right )}\right )+c_{1} \\ y \left (x \right ) &= -a \ln \left (x +a +\sqrt {x \left (2 a +x \right )}\right )+c_{1} \\ \end{align*}

Problem 4417

ODE	\[ \boxed {y-x y^{\prime }-y^{\prime }+{y^{\prime }}^{2}=0} \]
program solution	\[ y = -c_{1}^{2}+c_{1} x +c_{1} \] Verified OK. \[ y = \frac {\left (x +1\right )^{2}}{4} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (1+x \right )^{2}}{4} \\ y \left (x \right ) &= c_{1} \left (-c_{1} +x +1\right ) \\ \end{align*}

Problem 4418

ODE	\[ \boxed {y-x y^{\prime }-\sqrt {b^{2}-{y^{\prime }}^{2} a^{2}}=0} \]
program solution	\[ y = c_{1} x +\sqrt {-a^{2} c_{1}^{2}+b^{2}} \] Verified OK. \[ y = \frac {\sqrt {\frac {a^{2} b^{2}}{a^{2}+x^{2}}}\, \sqrt {a^{2}+x^{2}}\, a +b \,x^{2}}{\sqrt {a^{2}+x^{2}}\, a} \] Verified OK. \[ y = \frac {\sqrt {\frac {a^{2} b^{2}}{a^{2}+x^{2}}}\, \sqrt {a^{2}+x^{2}}\, a -b \,x^{2}}{\sqrt {a^{2}+x^{2}}\, a} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x +\sqrt {-a^{2} c_{1}^{2}+b^{2}} \]

Problem 4419

ODE	\[ \boxed {y-x y^{\prime }-x \sqrt {1+{y^{\prime }}^{2}}=0} \]
program solution	\[ y = -i x \] Verified OK. \[ y = i x \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (\sqrt {-\frac {c_{1}^{2}}{x \left (-2 c_{1} +x \right )}}\, \sqrt {-x \left (-2 c_{1} +x \right )}-x +c_{1} \right ) x}{\sqrt {-x \left (-2 c_{1} +x \right )}} \\ y \left (x \right ) &= \frac {\left (\sqrt {-\frac {c_{1}^{2}}{x \left (-2 c_{1} +x \right )}}\, \sqrt {-x \left (-2 c_{1} +x \right )}+x -c_{1} \right ) x}{\sqrt {-x \left (-2 c_{1} +x \right )}} \\ \end{align*}

Problem 4420

ODE	\[ \boxed {y-x y^{\prime }-a x \sqrt {1+{y^{\prime }}^{2}}=0} \]
program solution	\[ y = -i x \] Verified OK. \[ y = i x \] Verified OK.
Maple solution	\begin{align*} \frac {x \sqrt {\frac {-x^{2} a^{2}+y \left (x \right )^{2} a^{2}+2 \sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a +y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-x^{2} a^{2}+y \left (x \right )^{2} a^{2}+2 \sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\ \frac {x \sqrt {\frac {-x^{2} a^{2}+y \left (x \right )^{2} a^{2}-2 \sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {-\sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a +y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-x^{2} a^{2}+y \left (x \right )^{2} a^{2}-2 \sqrt {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\ \end{align*}

Problem 4421

ODE	\[ \boxed {-y^{\prime } y-a {y^{\prime }}^{2}=-x} \]
program solution	\[ y = a -x \] Verified OK. \[ y = -a +x \] Verified OK. \[ x = \left (-y+\sqrt {y^{2}+4 a x}\right ) \left (\frac {\sqrt {2}\, \left (\ln \left (2\right )-\ln \left (\frac {\sqrt {2}\, \sqrt {\frac {-y \sqrt {y^{2}+4 a x}-2 a^{2}+2 a x +y^{2}}{a^{2}}}\, a +\sqrt {y^{2}+4 a x}-y}{a}\right )\right )}{2 \sqrt {\frac {-y \sqrt {y^{2}+4 a x}-2 a^{2}+2 a x +y^{2}}{a^{2}}}}+\frac {c_{1}}{\sqrt {\frac {-y+\sqrt {y^{2}+4 a x}-2 a}{a}}\, \sqrt {\frac {-y+\sqrt {y^{2}+4 a x}+2 a}{a}}\, a}\right ) \] Warning, solution could not be verified \[ x = \left (y+\sqrt {y^{2}+4 a x}\right ) \left (-\frac {\sqrt {2}\, \left (\ln \left (2\right )-\ln \left (\frac {\sqrt {2}\, \sqrt {\frac {y \sqrt {y^{2}+4 a x}-2 a^{2}+2 a x +y^{2}}{a^{2}}}\, a -\sqrt {y^{2}+4 a x}-y}{a}\right )\right )}{2 \sqrt {\frac {y \sqrt {y^{2}+4 a x}-2 a^{2}+2 a x +y^{2}}{a^{2}}}}-\frac {c_{1}}{\sqrt {\frac {-y-\sqrt {y^{2}+4 a x}-2 a}{a}}\, \sqrt {\frac {-y-\sqrt {y^{2}+4 a x}+2 a}{a}}\, a}\right ) \] Warning, solution could not be verified
Maple solution	\begin{align*} \frac {c_{1} \left (y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}\right )}{\sqrt {\frac {-y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}-2 a}{a}}\, \sqrt {\frac {-y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}+2 a}{a}}}+x -\frac {\left (y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}\right ) \left (-3 \ln \left (2\right )+2 \ln \left (\frac {2 \sqrt {\frac {y \left (x \right )^{2}-y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}-2 a^{2}+2 a x}{a^{2}}}\, a -\left (y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}\right ) \sqrt {2}}{a}\right )\right ) \sqrt {2}}{4 \sqrt {\frac {y \left (x \right )^{2}-y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}-2 a^{2}+2 a x}{a^{2}}}} &= 0 \\ \frac {c_{1} \left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right )}{2 \sqrt {\frac {-y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}-2 a}{a}}\, \sqrt {\frac {-y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}+2 a}{a}}}+x -\frac {\left (-\frac {3 \ln \left (2\right )}{2}+\ln \left (\frac {2 \sqrt {\frac {y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}-2 a^{2}+2 a x +y \left (x \right )^{2}}{a^{2}}}\, a -\left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right ) \sqrt {2}}{a}\right )\right ) \left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right ) \sqrt {2}}{2 \sqrt {\frac {y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}-2 a^{2}+2 a x +y \left (x \right )^{2}}{a^{2}}}} &= 0 \\ \end{align*}

Problem 4422

ODE	\[ \boxed {y^{\prime } y-a \sqrt {1+{y^{\prime }}^{2}}=-x} \]
program solution	\[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ x = \frac {a \left (y x +\sqrt {-a^{2} \left (-y^{2}+a^{2}-x^{2}\right )}\right ) \arctan \left (\frac {y x +\sqrt {-a^{2} \left (-y^{2}+a^{2}-x^{2}\right )}}{a^{2}-y^{2}}\right )+a^{3}-y^{2} a +y c_{1} x +\sqrt {-a^{2} \left (-y^{2}+a^{2}-x^{2}\right )}\, c_{1}}{\sqrt {\frac {2 \sqrt {-a^{2} \left (-y^{2}+a^{2}-x^{2}\right )}\, x y+y^{4}+\left (-a^{2}+x^{2}\right ) y^{2}+a^{2} x^{2}}{\left (a^{2}-y^{2}\right )^{2}}}\, \left (a^{2}-y^{2}\right )} \] Verified OK. \[ x = \frac {-a \left (y x -\sqrt {-a^{2} \left (-y^{2}+a^{2}-x^{2}\right )}\right ) \arctan \left (\frac {-y x +\sqrt {-a^{2} \left (-y^{2}+a^{2}-x^{2}\right )}}{a^{2}-y^{2}}\right )+a^{3}-y^{2} a +y c_{1} x -\sqrt {-a^{2} \left (-y^{2}+a^{2}-x^{2}\right )}\, c_{1}}{\sqrt {\frac {-2 \sqrt {-a^{2} \left (-y^{2}+a^{2}-x^{2}\right )}\, x y+y^{4}+\left (-a^{2}+x^{2}\right ) y^{2}+a^{2} x^{2}}{\left (a^{2}-y^{2}\right )^{2}}}\, \left (a^{2}-y^{2}\right )} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \csc \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a -x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a -x \right )\right )\right ) \operatorname {csgn}\left (\sec \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a -x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a -x \right )\right )\right )\right ) a -\cot \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a -x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a -x \right )\right )\right ) x \\ y \left (x \right ) &= \csc \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a +x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a +x \right )\right )\right ) a \,\operatorname {csgn}\left (\sec \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a +x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a +x \right )\right )\right )\right )-\cot \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a +x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a +x \right )\right )\right ) x \\ \end{align*}

Problem 4423

ODE	\[ \boxed {y^{\prime } y-y^{2}+y^{2} {y^{\prime }}^{2}=x} \]
program solution
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-1-4 x}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-1-4 x}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {4 x^{2}+\left (-8 c_{1} -4\right ) x +4 c_{1}^{2}-1}}{2} \\ y \left (x \right ) &= \frac {\sqrt {4 x^{2}+\left (-8 c_{1} -4\right ) x +4 c_{1}^{2}-1}}{2} \\ \end{align*}

Problem 4424

ODE	\[ \boxed {y-\frac {1}{\sqrt {1+{y^{\prime }}^{2}}}-\frac {y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}}=x} \]
program solution	\[ y = x +\sqrt {2} \] Verified OK. \[ x = \frac {\left (-1+\left (-y+x \right ) \sqrt {-y^{2}+2 y x -x^{2}+2}\right ) \sqrt {2}}{2 \sqrt {\frac {-x \sqrt {-y^{2}+2 y x -x^{2}+2}+y \sqrt {-y^{2}+2 y x -x^{2}+2}+1}{\left (x^{2}-2 y x +y^{2}-1\right )^{2}}}\, \left (x -y+1\right ) \left (x -y-1\right )}+c_{2} \] Warning, solution could not be verified \[ x = -\frac {\sqrt {2}\, \left (1+\left (-y+x \right ) \sqrt {-y^{2}+2 y x -x^{2}+2}\right )}{2 \sqrt {\frac {x \sqrt {-y^{2}+2 y x -x^{2}+2}-y \sqrt {-y^{2}+2 y x -x^{2}+2}+1}{\left (x^{2}-2 y x +y^{2}-1\right )^{2}}}\, \left (x -y+1\right ) \left (x -y-1\right )}+c_{2} \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sqrt {-\frac {1}{\left (-c_{1} +x +1\right ) \left (x -c_{1} -1\right )}}+1}{\sqrt {-\frac {1}{\left (-c_{1} +x +1\right ) \left (x -c_{1} -1\right )}}} \]

Problem 4425

ODE	\[ \boxed {y-2 x y^{\prime }-x {y^{\prime }}^{2}=0} \]
program solution	\[ y = -x \] Verified OK. \[ y = 0 \] Verified OK. \[ y = \left (\frac {c_{1}^{2}}{x}+\frac {2 c_{1}}{\sqrt {x}}\right ) x \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -x \\ y \left (x \right ) &= c_{1} +2 \sqrt {c_{1} x} \\ y \left (x \right ) &= c_{1} -2 \sqrt {c_{1} x} \\ \end{align*}

Problem 4426

ODE	\[ \boxed {\frac {-x y^{\prime }+y}{y^{2}+y^{\prime }}-\frac {-x y^{\prime }+y}{1+y^{\prime } x^{2}}=0} \]
program solution	\[ y = -\tanh \left (-\operatorname {arctanh}\left (x \right )+c_{2} \right ) \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} x \\ y \left (x \right ) &= -\tanh \left (-\operatorname {arctanh}\left (x \right )+c_{1} \right ) \\ \end{align*}

Problem 4427

ODE	\[ \boxed {2 y x +\left (x^{2}+y^{2}\right ) y^{\prime }=0} \]
program solution	\[ y x^{2}+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}}}{4}\right )}{\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}{4 \sqrt {c_{1}}}-\frac {\sqrt {c_{1}}\, \left (i \sqrt {3}-1\right ) x^{2}}{\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}} \sqrt {3}+4 c_{1} x^{2}-\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}}}{4 \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ \end{align*}

Problem 4428

ODE	\[ \boxed {\left (x +\sqrt {y^{2}-y x}\right ) y^{\prime }-y=0} \]
program solution	\[ \frac {2 i \sqrt {-y+x}+\ln \left (y\right ) \sqrt {y}-c_{1} \sqrt {y}}{\sqrt {y}} = 0 \] Verified OK.
Maple solution	\[ \frac {\ln \left (y \left (x \right )\right ) y \left (x \right )-c_{1} y \left (x \right )+2 \sqrt {y \left (x \right ) \left (y \left (x \right )-x \right )}}{y \left (x \right )} = 0 \]

Problem 4429

ODE	\[ \boxed {y-y^{\prime } \left (-y+x \right )=-x} \]
program solution	\[ \frac {\ln \left (x^{2}+y^{2}\right )}{2}+\arctan \left (\frac {x}{y}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x \right )+2 c_{1} \right )\right ) x \]

Problem 4430

ODE	\[ \boxed {x y^{\prime }-y-x \sin \left (\frac {y}{x}\right )=0} \]
program solution	\[ y = \arctan \left (-\frac {2 c_{1} x}{c_{1}^{2} x^{2}+1}, -\frac {c_{1}^{2} x^{2}-1}{c_{1}^{2} x^{2}+1}\right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \arctan \left (\frac {2 x c_{1}}{x^{2} c_{1}^{2}+1}, \frac {-x^{2} c_{1}^{2}+1}{x^{2} c_{1}^{2}+1}\right ) x \]

Problem 4431

ODE	\[ \boxed {2 y x^{2}+y^{3}+\left (x y^{2}-2 x^{3}\right ) y^{\prime }=0} \]
program solution	\[ y = \frac {{\mathrm e}^{\frac {\operatorname {LambertW}\left (-2 x^{4} {\mathrm e}^{-4 c_{1}}\right )}{2}+2 c_{1}}}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {2}\, \sqrt {-\frac {1}{\operatorname {LambertW}\left (-2 c_{1} x^{4}\right )}}\, x \]

Problem 4432

ODE	\[ \boxed {y^{2}+\left (x \sqrt {y^{2}-x^{2}}-y x \right ) y^{\prime }=0} \]
program solution	\[ \ln \left (y\right )-\ln \left (\sqrt {y^{2}-x^{2}}+y\right ) = -\ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\[ \frac {-c_{1} x y \left (x \right )+y \left (x \right )+\sqrt {y \left (x \right )^{2}-x^{2}}}{x y \left (x \right )} = 0 \]

Problem 4433

ODE	\[ \boxed {\frac {y \cos \left (\frac {y}{x}\right )}{x}-\left (\frac {x \sin \left (\frac {y}{x}\right )}{y}+\cos \left (\frac {y}{x}\right )\right ) y^{\prime }=0} \]
program solution	\[ y \sin \left (\frac {y}{x}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (\textit {\_Z} x c_{1} \sin \left (\textit {\_Z} \right )-1\right ) x \]

Problem 4434

ODE	\[ \boxed {y+x \ln \left (\frac {y}{x}\right ) y^{\prime }-2 x y^{\prime }=0} \]
program solution	\[ y = \frac {\operatorname {LambertW}\left (c_{1} x \,{\mathrm e}\right )}{c_{1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\operatorname {LambertW}\left (-{\mathrm e} x c_{1} \right )}{c_{1}} \]

Problem 4435

ODE	\[ \boxed {2 y \,{\mathrm e}^{\frac {x}{y}}+\left (y-2 x \,{\mathrm e}^{\frac {x}{y}}\right ) y^{\prime }=0} \]
program solution	\[ \ln \left (y\right )+2 \,{\mathrm e}^{\frac {x}{y}} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x}{\operatorname {RootOf}\left (-\textit {\_Z} \,{\mathrm e}^{-2 \,{\mathrm e}^{\textit {\_Z}}}+c_{1} x \right )} \]

Problem 4436

ODE	\[ \boxed {x \,{\mathrm e}^{\frac {y}{x}}-y \sin \left (\frac {y}{x}\right )+x \sin \left (\frac {y}{x}\right ) y^{\prime }=0} \]
program solution	\[ -\frac {\left (\cos \left (\frac {y}{x}\right )+\sin \left (\frac {y}{x}\right )\right ) {\mathrm e}^{-\frac {y}{x}}}{2} = -\ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left ({\mathrm e}^{2 \textit {\_Z}} \left (4 \ln \left (x \right )^{2} {\mathrm e}^{2 \textit {\_Z}}+8 \ln \left (x \right ) {\mathrm e}^{2 \textit {\_Z}} c_{1} +4 \,{\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-4 \ln \left (x \right ) \sin \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z}}-4 \sin \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z}} c_{1} +2 \sin \left (\textit {\_Z} \right )^{2}-1\right )\right ) x \]

Problem 4437

ODE	\[ \boxed {y^{2}-2 x y y^{\prime }=-x^{2}} \] With initial conditions \begin {align*} [y \left (-1\right ) = 0] \end {align*}
program solution	\[ x -\frac {y^{2}}{x} = -1 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {x \left (1+x \right )} \\ y \left (x \right ) &= -\sqrt {x \left (1+x \right )} \\ \end{align*}

Problem 4438

ODE	\[ \boxed {x \,{\mathrm e}^{\frac {y}{x}}+y-x y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ y = -\ln \left (1+\ln \left (\frac {1}{x}\right )\right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \ln \left (-\frac {1}{\ln \left (x \right )-1}\right ) x \]

Problem 4439

ODE	\[ \boxed {y^{\prime }-\frac {y}{x}+\csc \left (\frac {y}{x}\right )=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ y = \frac {\pi x}{2}+x \arcsin \left (-1+\ln \left (\frac {1}{x}\right )\right ) \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \arccos \left (\ln \left (x \right )+1\right ) x \\ y \left (x \right ) &= -\arccos \left (\ln \left (x \right )+1\right ) x \\ \end{align*}

Problem 4440

ODE	\[ \boxed {y x -y^{2}-y^{\prime } x^{2}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
program solution	\[ y = \frac {x}{\ln \left (x \right )+1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x}{\ln \left (x \right )+1} \]

Problem 4441

ODE	\[ \boxed {2 y-\left (2 x -4 y\right ) y^{\prime }=-x +4} \]
program solution	\[ \frac {\ln \left (4 y^{2}+x^{2}-8 y-4 x +8\right )}{2}-\arctan \left (\frac {2 y-2}{x -2}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-\frac {\tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x -2\right )+2 c_{1} \right )\right ) \left (x -2\right )}{2} \]

Problem 4442

ODE	\[ \boxed {2 y-\left (3 x +2 y-1\right ) y^{\prime }=-3 x -1} \]
program solution	\[ y = -\frac {3 x}{2}-\frac {2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {25 x}{4}+\frac {1}{4}-\frac {25 c_{1}}{2}}}{4}\right )}{5}+\frac {1}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {3 x}{2}-\frac {2 \operatorname {LambertW}\left (-\frac {c_{1} {\mathrm e}^{\frac {1}{4}-\frac {25 x}{4}}}{4}\right )}{5}+\frac {1}{10} \]

Problem 4443

ODE	\[ \boxed {y+\left (2 x +2 y+2\right ) y^{\prime }=-x -1} \]
program solution	\[ y = -\frac {x}{2}+c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -1-x \\ y \left (x \right ) &= -\frac {x}{2}+c_{1} \\ \end{align*}

Problem 4444

ODE	\[ \boxed {y+\left (2 x +2 y-3\right ) y^{\prime }=1-x} \]
program solution	\[ y = \frac {\operatorname {LambertW}\left (2 \,{\mathrm e}^{x -4+c_{1}}\right )}{2}-x +2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {LambertW}\left (2 \,{\mathrm e}^{x -4-c_{1}}\right )}{2}+2-x \]

Problem 4445

ODE	\[ \boxed {y-\left (x -y-1\right ) y^{\prime }=1-x} \]
program solution	\[ -\frac {\ln \left (y^{2}+x^{2}-2 x +1\right )}{2}+\arctan \left (\frac {y}{-1+x}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x -1\right )+2 c_{1} \right )\right ) \left (1-x \right ) \]

Problem 4446

ODE	\[ \boxed {y+\left (2 x +2 y-1\right ) y^{\prime }=-x} \]
program solution	\[ y = \frac {\operatorname {LambertW}\left (2 \,{\mathrm e}^{x -2+c_{1}}\right )}{2}-x +1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {LambertW}\left (2 \,{\mathrm e}^{x -2-c_{1}}\right )}{2}-x +1 \]

Problem 4447

ODE	\[ \boxed {7 y+\left (1+2 x \right ) y^{\prime }=3} \]
program solution	\[ y = \frac {{\mathrm e}^{-7 c_{1} -\frac {7 \ln \left (1+2 x \right )}{2}}}{7}+\frac {3}{7} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3}{7}+\frac {c_{1}}{\left (1+2 x \right )^{\frac {7}{2}}} \]

Problem 4448

ODE	\[ \boxed {2 y+\left (3 x +6 y+3\right ) y^{\prime }=-x} \]
program solution	\[ y = -\frac {x}{2}-\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {x}{6}-\frac {c_{1}}{3}-\frac {3}{2}}}{2}\right )-\frac {3}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {3}{2}-\frac {x}{6}+\frac {c_{1}}{6}}}{2}\right )-\frac {3}{2}-\frac {x}{2} \]

Problem 4449

ODE	\[ \boxed {2 y+\left (y-1\right ) y^{\prime }=-x} \]
program solution	\[ y = {\mathrm e}^{\operatorname {LambertW}\left (-\left (x +2\right ) {\mathrm e}^{-c_{1}}\right )+c_{1}}-x -1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-1-x \right ) \operatorname {LambertW}\left (c_{1} \left (2+x \right )\right )-2-x}{\operatorname {LambertW}\left (c_{1} \left (2+x \right )\right )} \]

Problem 4450

ODE	\[ \boxed {-2 y-\left (2 x +7 y-1\right ) y^{\prime }=-3 x -4} \]
program solution	\[ \frac {x \left (3 x -4 y+8\right )}{2}+y-\frac {7 y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-\sqrt {7+15625 \left (x +\frac {26}{25}\right )^{2} c_{1}^{2}}+\left (-50 x +25\right ) c_{1}}{175 c_{1}} \]

Problem 4451

ODE	\[ \boxed {y+\left (3 x +3 y-4\right ) y^{\prime }=-x} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ y = \frac {2 \operatorname {LambertW}\left (-1, -\frac {3 \,{\mathrm e}^{x -\frac {5}{2}}}{2}\right )}{3}-x +2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 \operatorname {LambertW}\left (-1, -\frac {3 \,{\mathrm e}^{-\frac {5}{2}+x}}{2}\right )}{3}+2-x \]

Problem 4452

ODE	\[ \boxed {2 y-\left (x +2 y-1\right ) y^{\prime }=-3 x -3} \]
program solution	\[ \frac {4 \ln \left (2 y-3 x -9\right )}{5}+\frac {\ln \left (2 x +2 y+1\right )}{5} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-2-x \right ) {\operatorname {RootOf}\left (-1+\left (16 c_{1} x^{5}+160 c_{1} x^{4}+640 c_{1} x^{3}+1280 c_{1} x^{2}+1280 c_{1} x +512 c_{1} \right ) \textit {\_Z}^{25}+\left (-80 c_{1} x^{5}-800 c_{1} x^{4}-3200 c_{1} x^{3}-6400 c_{1} x^{2}-6400 c_{1} x -2560 c_{1} \right ) \textit {\_Z}^{20}\right )}^{5}}{2}+\frac {3 x}{2}+\frac {9}{2} \]

Problem 4453

ODE	\[ \boxed {y+\left (2 x +y+3\right ) y^{\prime }=-7} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ \left (y+7\right )^{2} x +\frac {y^{3}}{3}+5 y^{2}+21 y = {\frac {79}{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-x^{3}+6 x^{2}-12 x +72+8 \sqrt {-2 x^{3}+12 x^{2}-24 x +80}\right )^{\frac {1}{3}}+\frac {\left (x -2\right )^{2}}{\left (-x^{3}+6 x^{2}-12 x +72+8 \sqrt {-2 x^{3}+12 x^{2}-24 x +80}\right )^{\frac {1}{3}}}-x -5 \]

Problem 4454

ODE	\[ \boxed {y-\left (x -y-4\right ) y^{\prime }=-x -2} \]
program solution	\[ \frac {\ln \left (y^{2}+x^{2}+6 y-2 x +10\right )}{2}-\arctan \left (\frac {y+3}{-1+x}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -3-\tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x -1\right )+2 c_{1} \right )\right ) \left (x -1\right ) \]

Problem 4455

ODE	\[ \boxed {3 y x^{2}+8 x y^{2}+\left (x^{3}+8 y x^{2}+12 y^{2}\right ) y^{\prime }=0} \]
program solution	\[ y x^{2} \left (4 y+x \right )+4 y^{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (9 x^{5}-27 c_{1} -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_{1} x^{6}-54 c_{1} x^{5}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {x^{3} \left (-3+4 x \right )}{6 \left (9 x^{5}-27 c_{1} -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_{1} x^{6}-54 c_{1} x^{5}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x^{2}}{3} \\ y \left (x \right ) &= \frac {\frac {\left (-i \sqrt {3}-1\right ) \left (9 x^{5}-27 c_{1} -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_{1} x^{6}-54 c_{1} x^{5}+81 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (-\left (9 x^{5}-27 c_{1} -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_{1} x^{6}-54 c_{1} x^{5}+81 c_{1}^{2}}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (-\frac {3}{4}+x \right ) x \right ) x^{2}}{3 \left (9 x^{5}-27 c_{1} -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_{1} x^{6}-54 c_{1} x^{5}+81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (-\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (9 x^{5}-27 c_{1} -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_{1} x^{6}-54 c_{1} x^{5}+81 c_{1}^{2}}\right )^{\frac {2}{3}}+\left (\left (9 x^{5}-27 c_{1} -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_{1} x^{6}-54 c_{1} x^{5}+81 c_{1}^{2}}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (-\frac {3}{4}+x \right ) x \right ) x^{2}}{3 \left (9 x^{5}-27 c_{1} -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_{1} x^{6}-54 c_{1} x^{5}+81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 4456

ODE	\[ \boxed {\frac {2 y x +1}{y}+\frac {\left (y-x \right ) y^{\prime }}{y^{2}}=0} \]
program solution	\[ y = {\mathrm e}^{-x^{2}+\operatorname {LambertW}\left (-x \,{\mathrm e}^{x^{2}-c_{1}}\right )+c_{1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{\operatorname {LambertW}\left (-{\mathrm e}^{x^{2}} c_{1} x \right )} \]

Problem 4457

ODE	\[ \boxed {2 y x +\left (x^{2}+y^{2}\right ) y^{\prime }=0} \]
program solution	\[ y x^{2}+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}}}{4}\right )}{\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}{4 \sqrt {c_{1}}}-\frac {\sqrt {c_{1}}\, \left (i \sqrt {3}-1\right ) x^{2}}{\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}} \sqrt {3}+4 c_{1} x^{2}-\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {2}{3}}}{4 \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ \end{align*}

Problem 4458

ODE	\[ \boxed {{\mathrm e}^{x} \sin \left (y\right )+{\mathrm e}^{-y}-\left (x \,{\mathrm e}^{-y}-{\mathrm e}^{x} \cos \left (y\right )\right ) y^{\prime }=0} \]
program solution	\[ {\mathrm e}^{x} \sin \left (y\right )+x \,{\mathrm e}^{-y} = c_{1} \] Verified OK.
Maple solution	\[ {\mathrm e}^{x} \sin \left (y \left (x \right )\right )+x \,{\mathrm e}^{-y \left (x \right )}+c_{1} = 0 \]

Problem 4459

ODE	\[ \boxed {\cos \left (y\right )-\left (x \sin \left (y\right )-y^{2}\right ) y^{\prime }=0} \]
program solution	\[ x \cos \left (y\right )+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\[ x +\frac {\sec \left (y \left (x \right )\right ) \left (y \left (x \right )^{3}-3 c_{1} \right )}{3} = 0 \]

Problem 4460

ODE	\[ \boxed {-2 y x +{\mathrm e}^{y}+\left (y-x^{2}+x \,{\mathrm e}^{y}\right ) y^{\prime }=-x} \]
program solution	\[ x \,{\mathrm e}^{y}-x^{2} \left (y-\frac {1}{2}\right )+\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\[ -y \left (x \right ) x^{2}+x \,{\mathrm e}^{y \left (x \right )}+\frac {x^{2}}{2}+\frac {y \left (x \right )^{2}}{2}+c_{1} = 0 \]

Problem 4461

ODE	\[ \boxed {y^{2}-\left (-2 y x +{\mathrm e}^{y}\right ) y^{\prime }=-x^{2}+x} \]
program solution	\[ \frac {x^{3}}{3}+x y^{2}-\frac {x^{2}}{2}-{\mathrm e}^{y} = c_{1} \] Verified OK.
Maple solution	\[ \frac {x^{3}}{3}+x y \left (x \right )^{2}-\frac {x^{2}}{2}-{\mathrm e}^{y \left (x \right )}+c_{1} = 0 \]

Problem 4462

ODE	\[ \boxed {y \cos \left (x \right )+\left (2 y+\sin \left (x \right )-\sin \left (y\right )\right ) y^{\prime }=-2 x} \]
program solution	\[ y \sin \left (x \right )+x^{2}+y^{2}+\cos \left (y\right ) = c_{1} \] Verified OK.
Maple solution	\[ \sin \left (x \right ) y \left (x \right )+x^{2}+y \left (x \right )^{2}+\cos \left (y \left (x \right )\right )+c_{1} = 0 \]

Problem 4463

ODE	\[ \boxed {x \sqrt {x^{2}+y^{2}}-\frac {x^{2} y y^{\prime }}{y-\sqrt {x^{2}+y^{2}}}=0} \]
program solution	\[ \frac {\left (x^{2}+y^{2}\right )^{\frac {3}{2}}}{3}+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\[ c_{1} +\left (x^{2}+y \left (x \right )^{2}\right )^{\frac {3}{2}}+y \left (x \right )^{3} = 0 \]

Problem 4464

ODE	\[ \boxed {y^{3}-\left (y^{2}+1-3 x y^{2}\right ) y^{\prime }=-4 x^{3}+\sin \left (x \right )} \]
program solution	\[ x^{4}+x y^{3}+\cos \left (x \right )-\frac {y^{3}}{3}-y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (2^{\frac {1}{3}} {\left (\left (-3 x^{4}-3 \cos \left (x \right )+\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}-3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {2}{3}}+6 x -2\right ) 2^{\frac {1}{3}}}{{\left (\left (-3 x^{4}-3 \cos \left (x \right )+\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}-3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {1}{3}} \left (6 x -2\right )} \\ y \left (x \right ) &= -\frac {\left (2^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) {\left (-\left (3 x^{4}+3 \cos \left (x \right )-\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}+3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {2}{3}}-6 \left (x -\frac {1}{3}\right ) \left (i \sqrt {3}-1\right )\right ) 2^{\frac {1}{3}}}{4 {\left (-\left (3 x^{4}+3 \cos \left (x \right )-\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}+3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {1}{3}} \left (3 x -1\right )} \\ y \left (x \right ) &= \frac {\left (2^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) {\left (-\left (3 x^{4}+3 \cos \left (x \right )-\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}+3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {2}{3}}-6 \left (x -\frac {1}{3}\right ) \left (1+i \sqrt {3}\right )\right ) 2^{\frac {1}{3}}}{4 {\left (-\left (3 x^{4}+3 \cos \left (x \right )-\sqrt {\frac {\left (27 x -9\right ) \cos \left (x \right )^{2}+54 \left (x -\frac {1}{3}\right ) \left (x^{4}+c_{1} \right ) \cos \left (x \right )+27 x^{9}-9 x^{8}+54 c_{1} x^{5}-18 c_{1} x^{4}+27 c_{1}^{2} x -9 c_{1}^{2}-4}{3 x -1}}+3 c_{1} \right ) \left (3 x -1\right )^{2}\right )}^{\frac {1}{3}} \left (3 x -1\right )} \\ \end{align*}

Problem 4465

ODE	\[ \boxed {{\mathrm e}^{x} \left (y^{3}+x y^{3}+1\right )+3 y^{2} \left (x \,{\mathrm e}^{x}-6\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ y^{3} {\mathrm e}^{x} x -6 y^{3}+{\mathrm e}^{x} = -5 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (i \sqrt {3}-1\right ) {\left (-\left ({\mathrm e}^{x}+5\right ) \left (x \,{\mathrm e}^{x}-6\right )^{2}\right )}^{\frac {1}{3}}}{2 x \,{\mathrm e}^{x}-12} \]

Problem 4466

ODE	\[ \boxed {\sin \left (x \right ) \cos \left (y\right )+\cos \left (x \right ) \sin \left (y\right ) y^{\prime }=0} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{4}\right ) = \frac {\pi }{4}\right ] \end {align*}
program solution	\[ \ln \left (\cos \left (x \right )\right )+\ln \left (\cos \left (y\right )\right ) = -\ln \left (2\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\pi }{2}-\arcsin \left (\frac {\sec \left (x \right )}{2}\right ) \]

Problem 4467

ODE	\[ \boxed {y^{2} {\mathrm e}^{x y^{2}}+\left (2 x y \,{\mathrm e}^{x y^{2}}-3 y^{2}\right ) y^{\prime }=-4 x^{3}} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ {\mathrm e}^{x y^{2}}+x^{4}-y^{3} = 2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (-{\mathrm e}^{x \,\textit {\_Z}^{2}}-x^{4}+\textit {\_Z}^{3}+2\right ) \]

Problem 4468

ODE	\[ \boxed {y^{2}+y-x y^{\prime }=0} \]
program solution	\[ y = -\frac {x \,{\mathrm e}^{c_{1}}}{{\mathrm e}^{c_{1}} x -1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x}{-x +c_{1}} \]

Problem 4469

ODE	\[ \boxed {y \sec \left (x \right )+y^{\prime } \sin \left (x \right )=0} \]
program solution	\[ y = \frac {{\mathrm e}^{-c_{1}}}{\tan \left (x \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \cot \left (x \right ) c_{1} \]

Problem 4470

ODE	\[ \boxed {-\sin \left (y\right )+\cos \left (y\right ) y^{\prime }=-{\mathrm e}^{x}} \]
program solution	\[ x +\sin \left (y\right ) {\mathrm e}^{-x} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\arcsin \left (\left (x +c_{1} \right ) {\mathrm e}^{x}\right ) \]

Problem 4471

ODE	\[ \boxed {y x +\left (x^{2}+1\right ) y^{\prime }=0} \]
program solution	\[ y = {\mathrm e}^{-\frac {\ln \left (x^{2}+1\right )}{2}-c_{1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{\sqrt {x^{2}+1}} \]

Problem 4472

ODE	\[ \boxed {y^{3}+x y^{2}+y+\left (x^{3}+y x^{2}+x \right ) y^{\prime }=0} \]
program solution	\[ \frac {-2 y x -y^{2}-1}{2 x^{2} y^{2}}-\frac {1}{2 y^{2}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{2}+1}{\left (\sqrt {x^{2}+1}\, \sqrt {\frac {-1+\left (x^{4}+x^{2}\right ) c_{1}}{x^{2} \left (x^{2}+1\right )}}-1\right ) x} \\ y \left (x \right ) &= \frac {-x^{2}-1}{\left (\sqrt {x^{2}+1}\, \sqrt {\frac {-1+\left (x^{4}+x^{2}\right ) c_{1}}{x^{2} \left (x^{2}+1\right )}}+1\right ) x} \\ \end{align*}

Problem 4473

ODE	\[ \boxed {3 y-x y^{\prime }=0} \]
program solution	\[ y = {\mathrm e}^{3 c_{1}} x^{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \]

Problem 4474

ODE	\[ \boxed {y-3 x y^{\prime }=0} \]
program solution	\[ y = {\mathrm e}^{\frac {\ln \left (x \right )}{3}+\frac {c_{1}}{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \]

Problem 4475

ODE	\[ \boxed {y \left (2 y^{3} x^{2}+3\right )+x \left (y^{3} x^{2}-1\right ) y^{\prime }=0} \]
program solution	\[ \frac {2 \ln \left (y\right )}{11}-\frac {5 \ln \left (4 y^{3} x^{2}+11\right )}{22} = \frac {6 \ln \left (x \right )}{11}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-\frac {11 c_{1}}{3}} x^{3}}{\operatorname {RootOf}\left (11 \,{\mathrm e}^{11 c_{1}} \textit {\_Z}^{15}-{\mathrm e}^{11 c_{1}} \textit {\_Z}^{11}+4 x^{11}\right )^{5}} \]

Problem 4476

ODE	\[ \boxed {2 y x +\left (x^{2}+y^{2}\right ) y^{\prime }=-x^{2}} \]
program solution	\[ \frac {x^{2} \left (x +3 y\right )}{3}+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4 \sqrt {c_{1}}}-\frac {\left (i \sqrt {3}-1\right ) \sqrt {c_{1}}\, x^{2}}{\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}} \sqrt {3}+4 c_{1} x^{2}-\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}}}{4 \left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ \end{align*}

Problem 4477

ODE	\[ \boxed {y \cos \left (x \right )+\left (y^{3}+\sin \left (x \right )\right ) y^{\prime }=-x^{2}} \]
program solution	\[ \frac {x^{3}}{3}+y \sin \left (x \right )+\frac {y^{4}}{4} = c_{1} \] Verified OK.
Maple solution	\[ \frac {x^{3}}{3}+\sin \left (x \right ) y \left (x \right )+\frac {y \left (x \right )^{4}}{4}+c_{1} = 0 \]

Problem 4478

ODE	\[ \boxed {y^{2}+x y y^{\prime }=-x^{2}-x} \]
program solution	\[ \frac {x^{4}}{4}+\frac {y^{2} x^{2}}{2}+\frac {x^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-18 x^{4}-24 x^{3}+36 c_{1}}}{6 x} \\ y \left (x \right ) &= \frac {\sqrt {-18 x^{4}-24 x^{3}+36 c_{1}}}{6 x} \\ \end{align*}

Problem 4479

ODE	\[ \boxed {-2 y x +{\mathrm e}^{y}+\left (y-x^{2}+x \,{\mathrm e}^{y}\right ) y^{\prime }=-x} \]
program solution	\[ x \,{\mathrm e}^{y}-x^{2} \left (y-\frac {1}{2}\right )+\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\[ -y \left (x \right ) x^{2}+x \,{\mathrm e}^{y \left (x \right )}+\frac {x^{2}}{2}+\frac {y \left (x \right )^{2}}{2}+c_{1} = 0 \]

Problem 4480

ODE	\[ \boxed {{\mathrm e}^{x} \sin \left (y\right )+{\mathrm e}^{-y}-\left (x \,{\mathrm e}^{-y}-{\mathrm e}^{x} \cos \left (y\right )\right ) y^{\prime }=0} \]
program solution	\[ {\mathrm e}^{x} \sin \left (y\right )+x \,{\mathrm e}^{-y} = c_{1} \] Verified OK.
Maple solution	\[ {\mathrm e}^{x} \sin \left (y \left (x \right )\right )+x \,{\mathrm e}^{-y \left (x \right )}+c_{1} = 0 \]

Problem 4481

ODE	\[ \boxed {-y^{2}-y-\left (x^{2}-y^{2}-x \right ) y^{\prime }=-x^{2}} \]
program solution	\[ x -\frac {\ln \left (-y+x \right )}{2}+\frac {\ln \left (y+x \right )}{2}-y = c_{1} \] Verified OK.
Maple solution	\[ 2 y \left (x \right )-\ln \left (y \left (x \right )+x \right )+\ln \left (y \left (x \right )-x \right )-2 x -c_{1} = 0 \]

Problem 4482

ODE	\[ \boxed {y^{2} x^{4}-y+\left (y^{4} x^{2}-x \right ) y^{\prime }=0} \]
program solution	\[ \frac {y x^{4}+3}{3 x y}+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\[ -\frac {x^{3}}{3}-\frac {1}{y \left (x \right ) x}-\frac {y \left (x \right )^{3}}{3}+c_{1} = 0 \]

Problem 4483

ODE	\[ \boxed {y \left (2 x +y^{3}\right )-x \left (2 x -y^{3}\right ) y^{\prime }=0} \]
program solution	\[ \frac {x \left (y^{3}+x \right )}{y^{2}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\frac {\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}{2}+\frac {2 c_{1}^{2}}{\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}+c_{1}}{3 x} \\ y \left (x \right ) &= \frac {\left (-i \sqrt {3}-1\right ) \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {2}{3}}+4 \left (i c_{1} \sqrt {3}-c_{1} +\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}\right ) c_{1}}{12 \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {2}{3}}+4 \left (-i c_{1} \sqrt {3}-c_{1} +\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}\right ) c_{1}}{12 \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}} x} \\ \end{align*}

Problem 4484

ODE	\[ \boxed {\arctan \left (y x \right )+\frac {y x -2 x y^{2}}{y^{2} x^{2}+1}+\frac {\left (x^{2}-2 y x^{2}\right ) y^{\prime }}{y^{2} x^{2}+1}=0} \]
program solution	\[ -\ln \left (y^{2} x^{2}+1\right )+x \arctan \left (y x \right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (x \textit {\_Z} -\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+c_{1} \right )\right )}{x} \]

Problem 4485

ODE	\[ \boxed {\left (y \,{\mathrm e}^{y}-x \,{\mathrm e}^{x}\right ) y^{\prime }=-{\mathrm e}^{x} \left (x +1\right )} \]
program solution	\[ x \,{\mathrm e}^{-y+x}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\[ x \,{\mathrm e}^{-y \left (x \right )+x}+\frac {y \left (x \right )^{2}}{2}+c_{1} = 0 \]

Problem 4486

ODE	\[ \boxed {\frac {y x +1}{y}+\frac {\left (-x +2 y\right ) y^{\prime }}{y^{2}}=0} \]
program solution	\[ y = {\mathrm e}^{\operatorname {LambertW}\left (-\frac {x \,{\mathrm e}^{\frac {x^{2}}{4}-\frac {c_{1}}{2}}}{2}\right )-\frac {x^{2}}{4}+\frac {c_{1}}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {x^{2}}{4}} c_{1} x}{2}\right )} \]

Problem 4487

ODE	\[ \boxed {y^{2}-3 y x +\left (y x -x^{2}\right ) y^{\prime }=2 x^{2}} \]
program solution	\[ -\frac {x^{4}}{2}-x^{3} y+\frac {y^{2} x^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {c_{1} x^{2}-\sqrt {2 c_{1}^{2} x^{4}+1}}{c_{1} x} \\ y \left (x \right ) &= \frac {c_{1} x^{2}+\sqrt {2 c_{1}^{2} x^{4}+1}}{c_{1} x} \\ \end{align*}

Problem 4488

ODE	\[ \boxed {y \left (y+2 x +1\right )-x \left (x +2 y-1\right ) y^{\prime }=0} \]
program solution	\[ \frac {3 x -3 y-3}{\left (y x \right )^{\frac {1}{3}}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {3 \,5^{\frac {1}{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}-160 c_{1} x +80 c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{40 c_{1}}+\frac {3 x 5^{\frac {2}{3}}}{40 {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}-160 c_{1} x +80 c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}+x -1 \\ y \left (x \right ) &= \frac {\frac {3 \,5^{\frac {1}{3}} \left (-i \sqrt {3}-1\right ) {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {2}{3}}}{80}+\frac {3 c_{1} \left (\frac {80 \left (x -1\right ) {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{3}+\left (i \sqrt {3}-1\right ) 5^{\frac {2}{3}} x \right )}{80}}{c_{1} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {3 \left (i \sqrt {3}-1\right ) 5^{\frac {1}{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {2}{3}}}{80}+\frac {3 \left (-\frac {80 \left (1-x \right ) {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{3}+\left (-i \sqrt {3}-1\right ) 5^{\frac {2}{3}} x \right ) c_{1}}{80}}{c_{1} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}} \\ \end{align*}

Problem 4489

ODE	\[ \boxed {y \left (2 x -y-1\right )+x \left (2 y-x -1\right ) y^{\prime }=0} \]
program solution	\[ \frac {3 y+3 x +3}{\left (y x \right )^{\frac {1}{3}}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {3 \,5^{\frac {1}{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1} -x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{40 c_{1}}+\frac {3 x 5^{\frac {2}{3}}}{40 {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1} -x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}-1-x \\ y \left (x \right ) &= \frac {\frac {3 \,5^{\frac {1}{3}} \left (-i \sqrt {3}-1\right ) {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (1+x \right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {2}{3}}}{80}+\frac {3 c_{1} \left (\frac {80 \left (-1-x \right ) {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (1+x \right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {1}{3}}}{3}+\left (i \sqrt {3}-1\right ) 5^{\frac {2}{3}} x \right )}{80}}{{\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (1+x \right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {\frac {3 \left (i \sqrt {3}-1\right ) 5^{\frac {1}{3}} {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (1+x \right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {2}{3}}}{80}+\frac {3 \left (-\frac {80 \left (1+x \right ) {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (1+x \right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {1}{3}}}{3}+\left (-i \sqrt {3}-1\right ) 5^{\frac {2}{3}} x \right ) c_{1}}{80}}{{\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (1+x \right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 4490

ODE	\[ \boxed {y^{2}+12 y x^{2}+\left (2 y x +4 x^{3}\right ) y^{\prime }=0} \]
program solution	\[ 4 x^{3} y+x y^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-2 x^{3}+\sqrt {4 x^{6}+c_{1} x}}{x} \\ y \left (x \right ) &= \frac {-2 x^{3}-\sqrt {4 x^{6}+c_{1} x}}{x} \\ \end{align*}

Problem 4491

ODE	\[ \boxed {3 \left (y+x \right )^{2}+x \left (3 y+2 x \right ) y^{\prime }=0} \]
program solution	\[ \frac {3 x^{4}}{4}+2 x^{3} y+\frac {3 y^{2} x^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-4 c_{1} x^{2}-\sqrt {-2 c_{1}^{2} x^{4}+6}}{6 c_{1} x} \\ y \left (x \right ) &= \frac {-4 c_{1} x^{2}+\sqrt {-2 c_{1}^{2} x^{4}+6}}{6 c_{1} x} \\ \end{align*}

Problem 4492

ODE	\[ \boxed {y-\left (x^{2}+y^{2}+x \right ) y^{\prime }=0} \]
program solution	\[ \arctan \left (\frac {x}{y}\right )-y = c_{1} \] Verified OK.
Maple solution	\[ \frac {{\mathrm e}^{-2 i y \left (x \right )} \left (i x +y \left (x \right )\right )+2 \left (i y \left (x \right )+x \right ) c_{1}}{2 i y \left (x \right )+2 x} = 0 \]

Problem 4493

ODE	\[ \boxed {2 y x +\left (x^{2}+y^{2}+a \right ) y^{\prime }=0} \]
program solution	\[ y x^{2}+\frac {y^{3}}{3}+a y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}-4 x^{2}-4 a}{2 \left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}+\left (x^{2}+a \right ) \left (i \sqrt {3}-1\right )}{\left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (x^{2}+a \right ) \left (1+i \sqrt {3}\right )}{\left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 4494

ODE	\[ \boxed {2 y x +\left (x^{2}+y^{2}+a \right ) y^{\prime }=-x^{2}-b} \]
program solution	\[ \frac {x^{3}}{3}+y x^{2}+b x +\frac {y^{3}}{3}+a y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-4 x^{2}-4 a +\left (-4 x^{3}-12 x b -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 x b c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}}{2 \left (-4 x^{3}-12 x b -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 x b c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (-4 x^{3}-12 x b -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 x b c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}+\left (x^{2}+a \right ) \left (i \sqrt {3}-1\right )}{\left (-4 x^{3}-12 x b -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 x b c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{3}-12 x b -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 x b c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (x^{2}+a \right ) \left (1+i \sqrt {3}\right )}{\left (-4 x^{3}-12 x b -12 c_{1} +4 \sqrt {5 x^{6}+6 \left (2 a +b \right ) x^{4}+6 c_{1} x^{3}+3 \left (4 a^{2}+3 b^{2}\right ) x^{2}+18 x b c_{1} +4 a^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 4495

ODE	\[ \boxed {x y^{\prime }+y=x^{3}} \]
program solution	\[ y = \frac {x^{4}+4 c_{1}}{4 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{4}+4 c_{1}}{4 x} \]

Problem 4496

ODE	\[ \boxed {y^{\prime }+a y=b} \]
program solution	\[ y = -\frac {\frac {{\mathrm e}^{-a x}}{c_{1}}-b}{a} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-a x} c_{1} a +b}{a} \]

Problem 4497

ODE	\[ \boxed {x y^{\prime }+y-\ln \left (x \right ) y^{2}=0} \]
program solution	\[ y = \frac {1}{-c_{3} x +\ln \left (x \right )+1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{1+c_{1} x +\ln \left (x \right )} \]

Problem 4498

ODE	\[ \boxed {x^{\prime }+2 y x={\mathrm e}^{-y^{2}}} \]
program solution	\[ x = {\mathrm e}^{-y^{2}} \left (y +c_{1} \right ) \] Verified OK.
Maple solution	\[ x \left (y \right ) = \left (y +c_{1} \right ) {\mathrm e}^{-y^{2}} \]

Problem 4499

ODE	\[ \boxed {r^{\prime }-\left (r+{\mathrm e}^{-\theta }\right ) \tan \left (\theta \right )=0} \]
program solution	\[ r = -\frac {\sin \left (\theta \right ) {\mathrm e}^{-\theta }+{\mathrm e}^{-\theta } \cos \left (\theta \right )-2 c_{1}}{2 \cos \left (\theta \right )} \] Verified OK.
Maple solution	\[ r \left (\theta \right ) = \frac {\left (-\tan \left (\theta \right )-1\right ) {\mathrm e}^{-\theta }}{2}+\sec \left (\theta \right ) c_{1} \]

Problem 4500

ODE	\[ \boxed {y^{\prime }-\frac {2 x y}{x^{2}+1}=1} \]
program solution	\[ \frac {y}{x^{2}+1}-\arctan \left (x \right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\arctan \left (x \right )+c_{1} \right ) \left (x^{2}+1\right ) \]