2.17.152 Problems 15101 to 15200

Problem 15101

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

Problem 15102

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

Problem 15103

ODE	\[ \boxed {y^{\frac {2}{5}}+{y^{\prime }}^{\frac {2}{5}}=a^{\frac {2}{5}}} \]
program solution	\[ \frac {5 y^{\frac {3}{5}}}{3 \left (-y^{\frac {2}{5}}+a^{\frac {2}{5}}\right )^{\frac {3}{2}}}-\frac {5 y^{\frac {1}{5}}}{\sqrt {-y^{\frac {2}{5}}+a^{\frac {2}{5}}}}+5 \arctan \left (\frac {y^{\frac {1}{5}}}{\sqrt {-y^{\frac {2}{5}}+a^{\frac {2}{5}}}}\right ) = x +c_{1} \] Verified OK.
Maple solution	\[ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\left (a^{\frac {2}{5}}-\textit {\_a}^{\frac {2}{5}}\right )^{\frac {5}{2}}}d \textit {\_a} \right )-c_{1} = 0 \]

Problem 15104

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

Problem 15105

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

Problem 15106

ODE	\[ \boxed {y-\arcsin \left (y^{\prime }\right )-\ln \left (1+{y^{\prime }}^{2}\right )=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\sin \left (\operatorname {RootOf}\left (-\textit {\_a} +\textit {\_Z} +\ln \left (\sin \left (\textit {\_Z} \right )^{2}+1\right )\right )\right )}d \textit {\_a} = x +c_{1} \] Warning, solution could not be verified
Maple solution	\[ x -\left (\int _{}^{y \left (x \right )}\csc \left (\operatorname {RootOf}\left (-\textit {\_a} +\textit {\_Z} +\ln \left (2-\cos \left (\textit {\_Z} \right )^{2}\right )\right )\right )d \textit {\_a} \right )-c_{1} = 0 \]

Problem 15107

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

Problem 15108

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

Problem 15109

ODE	\[ \boxed {y-2 y^{\prime } x -\sin \left (y^{\prime }\right )=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {-\cos \left (\operatorname {RootOf}\left (2 \textit {\_Z} x +\sin \left (\textit {\_Z} \right )-y\right )\right )-\operatorname {RootOf}\left (2 \textit {\_Z} x +\sin \left (\textit {\_Z} \right )-y\right ) \sin \left (\operatorname {RootOf}\left (2 \textit {\_Z} x +\sin \left (\textit {\_Z} \right )-y\right )\right )+c_{1}}{\operatorname {RootOf}\left (2 \textit {\_Z} x +\sin \left (\textit {\_Z} \right )-y\right )^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ \left [x \left (\textit {\_T} \right ) &= \frac {-\textit {\_T} \sin \left (\textit {\_T} \right )-\cos \left (\textit {\_T} \right )+c_{1}}{\textit {\_T}^{2}}, y \left (\textit {\_T} \right ) &= \frac {-\textit {\_T} \sin \left (\textit {\_T} \right )-2 \cos \left (\textit {\_T} \right )+2 c_{1}}{\textit {\_T}}\right ] \\ \end{align*}

Problem 15110

ODE	\[ \boxed {y-x {y^{\prime }}^{2}+\frac {1}{y^{\prime }}=0} \]
program solution	\[ y = x -1 \] Verified OK. \[ x = -\frac {54 x^{3} \left (-\frac {x 2^{\frac {1}{3}} \left (\sqrt {\frac {-4 y^{3}+27 x}{x}}\, c_{1} 3^{\frac {1}{6}}+2 \,3^{\frac {2}{3}} \left (y+\frac {3 c_{1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{3}-\frac {2^{\frac {2}{3}} 3^{\frac {5}{6}} x^{2} \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}-3 x \left (\frac {2 y^{2} c_{1}}{9}+x \right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}}+\left (-\frac {4 y c_{1}}{3}+x \right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{\left (y 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2} \left (2 y 2^{\frac {1}{3}} 3^{\frac {2}{3}} x +2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}-6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be verified \[ x = -\frac {36 x^{3} \left (\left (\frac {8 y c_{1}}{9}-\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {2^{\frac {1}{3}} \left (c_{1} \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) \left (y+\frac {3 c_{1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (\frac {2 y^{2} c_{1}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{{\left (-\frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} y\right )\right )}^{2} {\left (\left (\sqrt {3}+i\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (i 3^{\frac {1}{3}}-3^{\frac {5}{6}}\right ) y 2^{\frac {2}{3}}\right )}^{2}} \] Warning, solution could not be verified \[ x = \frac {36 x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\left (-\frac {8 y c_{1}}{9}+\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {2^{\frac {1}{3}} \left (c_{1} \left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) \left (y+\frac {3 c_{1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (\frac {2 y^{2} c_{1}}{9}+x \right ) \left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right )\right ) 2^{\frac {2}{3}}\right )\right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{{\left (\left (i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) y 2^{\frac {2}{3}}\right )}^{2} {\left (-\frac {\left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) y\right )\right )}^{2}} \] Warning, solution could not be verified
Maple solution	\begin{align*} \frac {12 x^{3} \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )+x \left (\frac {2^{\frac {1}{3}} \left (3^{\frac {1}{6}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+3 \,3^{\frac {2}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{2}+2^{\frac {2}{3}} 3^{\frac {1}{3}} y \left (x \right )^{2}\right )\right ) c_{1} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{\left (2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}+2 x \left (2^{\frac {1}{3}} 3^{\frac {2}{3}} y \left (x \right )-3 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )\right )^{2} \left (y \left (x \right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}}+x -\frac {18 x^{4} \left (2^{\frac {2}{3}} 3^{\frac {5}{6}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}\, x +2 y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+9 \,3^{\frac {1}{3}} 2^{\frac {2}{3}} x -3 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{\left (2 y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}} x +2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}-6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2} \left (y \left (x \right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}} &= 0 \\ -\frac {3 x^{3} \left (\frac {8 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )}{9}+x \left (\left (\left (\frac {i 3^{\frac {2}{3}}}{9}-\frac {3^{\frac {1}{6}}}{9}\right ) \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}-\frac {2 \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} y \left (x \right )^{2}}{9}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} c_{1}}{2 {\left (\left (i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}} y \left (x \right ) x \right )}^{2} {\left (-\frac {\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) y \left (x \right ) 2^{\frac {1}{3}}\right )\right )}^{2}}+x +\frac {216 x^{4} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (-{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y \left (x \right ) \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\frac {x \left (-i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) 2^{\frac {2}{3}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}}{2}+\frac {3 x \left (-i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{2}\right )}{{\left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (-i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) x y \left (x \right ) 2^{\frac {2}{3}}\right )}^{2} {\left (\frac {\left (-3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{2}+x \left (6 i {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {2}{3}}+3 \,3^{\frac {1}{6}}\right ) y \left (x \right ) 2^{\frac {1}{3}}\right )\right )}^{2}} &= 0 \\ \frac {3 x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} c_{1} \left (-\frac {8 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )}{9}+x \left (\left (\left (\frac {i 3^{\frac {2}{3}}}{9}+\frac {3^{\frac {1}{6}}}{9}\right ) \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}-\frac {2 \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} y \left (x \right )^{2}}{9}\right )\right )}{2 {\left (-\frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{6}+\left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} y \left (x \right ) \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2} {\left (\left (1-i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) y \left (x \right ) x \right )}^{2}}+x +\frac {216 x^{4} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}} 3^{\frac {1}{3}} 2^{\frac {2}{3}} \left ({\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y \left (x \right ) \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\frac {x \left (-i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) 2^{\frac {2}{3}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}}{2}+\frac {3 x \left (-i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{2}\right )}{\left (\frac {2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}} \left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right )}{2}+\left (6 i {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+y \left (x \right ) \left (i 3^{\frac {2}{3}}-3 \,3^{\frac {1}{6}}\right ) 2^{\frac {1}{3}}\right ) x \right )^{2} {\left ({\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\sqrt {3}+i\right )+\left (-3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) x y \left (x \right ) 2^{\frac {2}{3}}\right )}^{2}} &= 0 \\ \end{align*}

Problem 15111

ODE	\[ \boxed {y-\frac {3 x y^{\prime }}{2}-{\mathrm e}^{y^{\prime }}=0} \]
program solution	\[ y = 1 \] Verified OK. \[ x = -\frac {27 \left (\left (-2 \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )^{2} x^{2}-4 x \left (x -\frac {2 y}{3}\right ) \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )-4 x^{2}+\frac {8 y x}{3}-\frac {8 y^{2}}{9}\right ) {\mathrm e}^{-\frac {3 x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )-2 y}{3 x}}+c_{1} x^{2}\right ) x}{{\left (3 x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )-2 y\right )}^{3}} \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= 1 \\ \frac {27 \left (\left (-2 x^{2} \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y \left (x \right )}{3 x}}}{3 x}\right )^{2}-4 \left (x -\frac {2 y \left (x \right )}{3}\right ) x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y \left (x \right )}{3 x}}}{3 x}\right )-4 x^{2}+\frac {8 x y \left (x \right )}{3}-\frac {8 y \left (x \right )^{2}}{9}\right ) {\mathrm e}^{\frac {-3 x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y \left (x \right )}{3 x}}}{3 x}\right )+2 y \left (x \right )}{3 x}}+\operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y \left (x \right )}{3 x}}}{3 x}\right )^{3} x^{3}-2 y \left (x \right ) \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y \left (x \right )}{3 x}}}{3 x}\right )^{2} x^{2}+\frac {4 y \left (x \right )^{2} \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y \left (x \right )}{3 x}}}{3 x}\right ) x}{3}+\frac {c_{1} x^{2}}{27}-\frac {8 y \left (x \right )^{3}}{27}\right ) x}{{\left (3 x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y \left (x \right )}{3 x}}}{3 x}\right )-2 y \left (x \right )\right )}^{3}} &= 0 \\ \end{align*}

Problem 15112

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

Problem 15113

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

Problem 15114

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

Problem 15115

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

Problem 15116

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

Problem 15117

ODE	\[ \boxed {y^{\prime } {\mathrm e}^{-x}+y^{2}-2 y \,{\mathrm e}^{x}=1-{\mathrm e}^{2 x}} \]
program solution	\[ y = \frac {\left (\left (\textit {\_C3} +1\right ) {\mathrm e}^{-x}+2 \,{\mathrm e}^{x}\right ) \cosh \left (\frac {x}{2}\right )+2 \left (\left (\frac {\textit {\_C3}}{2}+\frac {1}{2}\right ) {\mathrm e}^{-x}+\textit {\_C3} \,{\mathrm e}^{x}\right ) \sinh \left (\frac {x}{2}\right )}{2 \textit {\_C3} \sinh \left (\frac {x}{2}\right )+2 \cosh \left (\frac {x}{2}\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{x}+{\mathrm e}^{2 x} c_{1} +c_{1}}{{\mathrm e}^{x} c_{1} +1} \]

Problem 15118

ODE	\[ \boxed {y^{\prime }+y^{2}-2 y \sin \left (x \right )=-\sin \left (x \right )^{2}+\cos \left (x \right )} \]
program solution	\[ y = \frac {\sin \left (x \right ) \left (\textit {\_C3} +x \right )+1}{\textit {\_C3} +x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (x \right )+\frac {1}{x -c_{1}} \]

Problem 15119

ODE	\[ \boxed {x y^{\prime }-y^{2}+\left (2 x +1\right ) y=x^{2}+2 x} \]
program solution	\[ y = \frac {\textit {\_C3} \,x^{2}+x +1}{\textit {\_C3} x +1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2}-x -1}{c_{1} x -1} \]

Problem 15120

ODE	\[ \boxed {x^{2} y^{\prime }-y^{2} x^{2}-y x=1} \]
program solution	\[ y = \frac {-\ln \left (x \right )-\textit {\_C3} -1}{x \left (\ln \left (x \right )+\textit {\_C3} \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-\ln \left (x \right )+c_{1} -1}{x \left (-c_{1} +\ln \left (x \right )\right )} \]

Problem 15121

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

Problem 15122

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

Problem 15123

ODE	\[ \boxed {{y^{\prime }}^{3}-4 x y y^{\prime }+8 y^{2}=0} \]
program solution	\[ \left (3 i 3^{\frac {2}{3}}+3 \,3^{\frac {1}{6}}\right ) \left (\int _{}^{\frac {y}{x^{3}}}-\frac {\left (3 \sqrt {3}\, \sqrt {\textit {\_a}}-\sqrt {27 \textit {\_a} -4}\right )^{\frac {1}{3}}}{\sqrt {\textit {\_a}}\, \left (-9 \sqrt {\textit {\_a}}\, \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \left (3 \sqrt {3}\, \sqrt {\textit {\_a}}-\sqrt {27 \textit {\_a} -4}\right )^{\frac {1}{3}}+3 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {2}{3}} \left (-3 \sqrt {3}\, \sqrt {\textit {\_a}}+\sqrt {27 \textit {\_a} -4}\right )^{\frac {2}{3}}+4 \,2^{\frac {1}{3}} 3^{\frac {2}{3}}\right )}d \textit {\_a} \right )-c_{1} +\ln \left (x \right ) = 0 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {4 x^{3}}{27} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (4 c_{1} x -1\right )^{2}}{64 c_{1}^{3}} \\ \end{align*}

Problem 15124

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

Problem 15125

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

Problem 15126

ODE	\[ \boxed {\left (x y^{\prime }+y\right )^{2}+3 x^{5} \left (x y^{\prime }-2 y\right )=0} \]
program solution	\[ \frac {\ln \left (y\right )}{6}+\frac {\ln \left (-x^{5}+x^{\frac {5}{2}} \sqrt {x^{5}+4 y}\right )}{6}-\frac {\ln \left (x^{5}+x^{\frac {5}{2}} \sqrt {x^{5}+4 y}\right )}{6} = -\frac {\ln \left (x \right )}{6}+c_{1} \] Verified OK. \[ \frac {\ln \left (y\right )}{6}-\frac {\ln \left (-x^{5}+x^{\frac {5}{2}} \sqrt {x^{5}+4 y}\right )}{6}+\frac {\ln \left (x^{5}+x^{\frac {5}{2}} \sqrt {x^{5}+4 y}\right )}{6} = -\frac {\ln \left (x \right )}{6}+c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {x^{5}}{4} \\ y \left (x \right ) &= \frac {c_{1} \left (x^{3}+c_{1} \right )}{x} \\ y \left (x \right ) &= \frac {c_{1} \left (-x^{3}+c_{1} \right )}{x} \\ y \left (x \right ) &= \frac {c_{1} \left (-x^{3}+c_{1} \right )}{x} \\ y \left (x \right ) &= \frac {c_{1} \left (x^{3}+c_{1} \right )}{x} \\ \end{align*}

Problem 15127

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

Problem 15128

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

Problem 15129

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

Problem 15130

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

Problem 15131

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

Problem 15132

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

Problem 15133

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

Problem 15134

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

Problem 15135

ODE	\[ \boxed {y-x y^{\prime }-\sqrt {a^{2} {y^{\prime }}^{2}+b^{2}}=0} \]
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 15136

ODE	\[ \boxed {y^{\prime }-\left (-y+x \right )^{2}=1} \]
program solution	\[ y = \frac {\textit {\_C3} x +x^{2}-1}{\textit {\_C3} +x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x +x^{2}-1}{x +c_{1}} \]

Problem 15137

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

Problem 15138

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

Problem 15139

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

Problem 15140

ODE	\[ \boxed {5 y x -4 y^{2}+\left (y^{2}-8 y x +\frac {5 x^{2}}{2}\right ) y^{\prime }=6 x^{2}} \]
program solution	\[ -2 x \left (x^{2}-\frac {5 y x}{4}+2 y^{2}\right )+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y &= \frac {\frac {\left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {27 x^{2} c_{1}^{2}}{\left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+4 c_{1} x}{c_{1}} \\ y &= \frac {54 i \sqrt {3}\, c_{1}^{2} x^{2}-i \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {2}{3}} \sqrt {3}-54 x^{2} c_{1}^{2}+16 c_{1} x \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}-\left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {2}{3}}}{4 \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}} c_{1}} \\ y &= -\frac {54 i \sqrt {3}\, c_{1}^{2} x^{2}-i \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {2}{3}} \sqrt {3}+54 x^{2} c_{1}^{2}-16 c_{1} x \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}+\left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {2}{3}}}{4 \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 15141

ODE	\[ \boxed {3 y^{2} x +\left (3 y x^{2}-6 y^{2}-1\right ) y^{\prime }=x^{2}} \]
program solution	\[ \frac {3 y^{2} x^{2}}{2}-\frac {x^{3}}{3}-2 y^{3}-y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y &= \frac {\left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+162 c_{1} x^{6}+144 x^{6}+216 x^{5}-864 c_{1} x^{3}-27 x^{4}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{12}+\frac {3 x^{4}-8}{4 \left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+162 c_{1} x^{6}+144 x^{6}+216 x^{5}-864 c_{1} x^{3}-27 x^{4}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4} \\ y &= \frac {24+i \left (-24+9 x^{4}-\left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}\right ) \sqrt {3}-9 x^{4}+6 x^{2} \left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}-\left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}}{24 \left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}} \\ y &= \frac {24+i \left (-9 x^{4}+\left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}+24\right ) \sqrt {3}-9 x^{4}+6 x^{2} \left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}-\left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}}{24 \left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 15142

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

Problem 15143

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

Problem 15144

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

Problem 15145

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

Problem 15146

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

Problem 15147

ODE	\[ \boxed {\frac {1}{x^{2}-y x +y^{2}}-\frac {y^{\prime }}{2 y^{2}-y x}=0} \]
program solution	\[ \frac {3 \ln \left (y-2 x \right )}{2}-\ln \left (y-x \right )+\frac {\ln \left (y\right )}{2} = c_{1} \] Verified OK.
Maple solution	\[ y = \left (\operatorname {RootOf}\left (\textit {\_Z}^{8} c_{1} x^{2}+2 \textit {\_Z}^{6} c_{1} x^{2}-\textit {\_Z}^{4}-2 \textit {\_Z}^{2}-1\right )^{2}+2\right ) x \]

Problem 15148

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

Problem 15149

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

Problem 15150

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

Problem 15151

ODE	\[ \boxed {y^{\prime } \left (3 x^{2}-2 x \right )-y \left (6 x -2\right )=0} \]
program solution	\[ y = {\mathrm e}^{2 c_{1}} x \left (3 x -2\right ) \] Verified OK.
Maple solution	\[ y = c_{1} x \left (3 x -2\right ) \]

Problem 15152

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

Problem 15153

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

Problem 15154

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

Problem 15155

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

Problem 15156

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

Problem 15157

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

Problem 15158

ODE	\[ \boxed {\left (x -1\right ) \left (y^{2}-y+1\right )-\left (y-1\right ) \left (x^{2}+x +1\right ) y^{\prime }=0} \]
program solution	\[ -\frac {\ln \left (x^{2}+x +1\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )+\frac {\ln \left (y^{2}-y+1\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 y-1\right ) \sqrt {3}}{3}\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 15159

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

Problem 15160

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

Problem 15161

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

Problem 15162

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

Problem 15163

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

Problem 15164

ODE	\[ \boxed {\left (3 x +3 y+a^{2}\right ) y^{\prime }-4 y=b^{2}+4 x} \]
program solution	\[ y = \frac {4 \operatorname {LambertW}\left (\frac {3 \,{\mathrm e}^{\frac {3 a^{2}+3 b^{2}+49 c_{1} +49 x}{4 a^{2}-3 b^{2}}}}{4 a^{2}-3 b^{2}}\right ) a^{2}}{21}-\frac {\operatorname {LambertW}\left (\frac {3 \,{\mathrm e}^{\frac {3 a^{2}+3 b^{2}+49 c_{1} +49 x}{4 a^{2}-3 b^{2}}}}{4 a^{2}-3 b^{2}}\right ) b^{2}}{7}-\frac {a^{2}}{7}-\frac {b^{2}}{7}-x \] Verified OK.
Maple solution	\[ y = \frac {\left (4 a^{2}-3 b^{2}\right ) \operatorname {LambertW}\left (\frac {3 \,{\mathrm e}^{\frac {3 a^{2}+3 b^{2}-49 c_{1} +49 x}{4 a^{2}-3 b^{2}}}}{4 a^{2}-3 b^{2}}\right )}{21}-\frac {a^{2}}{7}-\frac {b^{2}}{7}-x \]

Problem 15165

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

Problem 15166

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

Problem 15167

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

Problem 15168

ODE	\[ \boxed {y^{\prime }-\sqrt {\frac {9 y^{2}-6 y+2}{x^{2}-2 x +5}}=0} \]
program solution	\[ -\frac {\sqrt {\frac {9 y^{2}-6 y+2}{x^{2}-2 x +5}}\, \sqrt {x^{2}-2 x +5}\, \operatorname {arcsinh}\left (\frac {x}{2}-\frac {1}{2}\right )}{\sqrt {9 y^{2}-6 y+2}}+\frac {\operatorname {arcsinh}\left (3 y-1\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ -\frac {\sqrt {\frac {9 y^{2}-6 y+2}{x^{2}-2 x +5}}\, \sqrt {x^{2}-2 x +5}\, \operatorname {arcsinh}\left (\frac {x}{2}-\frac {1}{2}\right )}{\sqrt {9 y^{2}-6 y+2}}+\frac {\operatorname {arcsinh}\left (3 y-1\right )}{3}+c_{1} = 0 \]

Problem 15169

ODE	\[ \boxed {\left (5 x -7 y+1\right ) y^{\prime }+y=1-x} \]
program solution	\[ \frac {\ln \left (y-x \right )}{8}+\frac {3 \ln \left (x +7 y-4\right )}{8} = c_{1} \] Verified OK.
Maple solution	\[ y = \frac {-\operatorname {RootOf}\left (7 \textit {\_Z}^{16}+\left (-128 c_{1} x^{4}+256 c_{1} x^{3}-192 c_{1} x^{2}+64 c_{1} x -8 c_{1} \right ) \textit {\_Z}^{4}-16 c_{1} x^{4}+32 c_{1} x^{3}-24 c_{1} x^{2}+8 c_{1} x -c_{1} \right )^{12}+16 c_{1} \left (x -\frac {1}{2}\right )^{3} x}{2 c_{1} \left (2 x -1\right )^{3}} \]

Problem 15170

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

Problem 15171

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

Problem 15172

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

Problem 15173

ODE	\[ \boxed {4 x^{2} {y^{\prime }}^{2}-y^{2}-y^{3} x=0} \]
program solution	\[ -\frac {8 \ln \left (\sqrt {y x +1}+2\right )}{3}+2 \ln \left (\sqrt {y x +1}+1\right )+\frac {2 \ln \left (\sqrt {y x +1}-1\right )}{3} = \ln \left (x \right )+c_{1} \] Verified OK. \[ \frac {2 \ln \left (\sqrt {y x +1}+1\right )}{3}-\frac {8 \ln \left (\sqrt {y x +1}-2\right )}{3}+2 \ln \left (\sqrt {y x +1}-1\right ) = \ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\begin{align*} y &= 0 \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

Problem 15174

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

Problem 15175

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

Problem 15176

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

Problem 15177

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

Problem 15178

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

Problem 15179

ODE	\[ \boxed {{y^{\prime }}^{4}=1} \]
program solution	\[ y = i x +c_{3} \] Verified OK. \[ y = -i x +\textit {\_C4} \] Verified OK.
Maple solution	\begin{align*} y &= -i x +c_{1} \\ y &= i x +c_{1} \\ y &= x +c_{1} \\ y &= -x +c_{1} \\ \end{align*}

Problem 15180

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

Problem 15181

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

Problem 15182

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

Problem 15183

ODE	\[ \boxed {{y^{\prime }}^{2}+y y^{\prime \prime }=1} \]
program solution	\[ \frac {y^{2}}{2}-\frac {x^{2}}{4}-\frac {c_{1} x}{2}-c_{2} = 0 \] Warning, solution could not be verified
Maple solution	\begin{align*} y &= \sqrt {-2 c_{1} x +x^{2}+2 c_{2}} \\ y &= -\sqrt {-2 c_{1} x +x^{2}+2 c_{2}} \\ \end{align*}

Problem 15184

ODE	\[ \boxed {y^{\prime \prime \prime \prime }=x} \]
program solution	\[ y = c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} +\frac {1}{120} x^{5} \] Verified OK.
Maple solution	\[ y = \frac {x^{5}}{120}+\frac {c_{1} x^{3}}{6}+\frac {c_{2} x^{2}}{2}+\frac {\left (3 c_{1}^{2}+2 c_{3} \right ) x}{2}+c_{4} \]

Problem 15185

ODE	\[ \boxed {y^{\prime \prime \prime }=x +\cos \left (x \right )} \]
program solution	\[ y = c_{3} x^{2}+c_{2} x +c_{1} +\frac {x^{4}}{24}-\sin \left (x \right ) \] Verified OK.
Maple solution	\[ y = \frac {x^{4}}{24}+\frac {c_{1} x^{2}}{2}-\sin \left (x \right )+c_{2} x +c_{3} \]

Problem 15186

ODE	\[ \boxed {y^{\prime \prime } \left (x +2\right )^{5}=1} \] With initial conditions \begin {align*} \left [y \left (-1\right ) = {\frac {1}{12}}, y^{\prime }\left (-1\right ) = -{\frac {1}{4}}\right ] \end {align*}
program solution	\[ y = \frac {1}{12 \left (x +2\right )^{3}} \] Verified OK.
Maple solution	\[ y = \frac {1}{12 \left (x +2\right )^{3}} \]

Problem 15187

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

Problem 15188

ODE	\[ \boxed {y^{\prime \prime }=2 \ln \left (x \right ) x} \]
program solution	\[ y = -\frac {5 x^{3}}{18}+\frac {\ln \left (x \right ) x^{3}}{3}+c_{1} x +c_{2} \] Verified OK.
Maple solution	\[ y = -\frac {5 x^{3}}{18}+\frac {x^{3} \ln \left (x \right )}{3}+c_{1} x +c_{2} \]

Problem 15189

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

Problem 15190

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

Problem 15191

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

Problem 15192

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

Problem 15193

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

Problem 15194

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

Problem 15195

ODE	\[ \boxed {2 y^{\prime \prime }-\frac {y^{\prime }}{x}-\frac {x^{2}}{y^{\prime }}=0} \] With initial conditions \begin {align*} \left [y \left (1\right ) = \frac {\sqrt {2}}{5}, y^{\prime }\left (1\right ) = \frac {\sqrt {2}}{2}\right ] \end {align*}
program solution	\[ y = \frac {x^{\frac {5}{2}} \sqrt {2}}{5} \] Verified OK. \[ y = -\frac {x^{\frac {5}{2}} \sqrt {2}}{5}+\frac {2 \sqrt {2}}{5} \] Warning, solution could not be verified
Maple solution	\[ y = \frac {\sqrt {2}\, x^{\frac {5}{2}}}{5} \]

Problem 15196

ODE	\[ \frac {d^{3}}{d x^{3}}y \left (x \right ) = \sqrt {1-\left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )^{2}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 15197

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

Problem 15198

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

Problem 15199

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

Problem 15200

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