2.17.101 Problems 10001 to 10100

Problem 10001

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

Problem 10002

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-x^{4} {y^{\prime }}^{2}+4 y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10003

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+a y^{3}+2 y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{2} \operatorname {JacobiSN}\left (\frac {\left (2 c_{1} x^{3}+\sqrt {2}\, \sqrt {x^{\frac {20}{3}} a}\right ) c_{2}}{2 x^{3}}, i\right ) x^{\frac {1}{3}} \]

Problem 10004

ODE	\[ \boxed {x^{3} \left (y^{\prime \prime }+y^{\prime } y-y^{3}\right )+12 x y=-24} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10005

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

Problem 10006

ODE	\[ \boxed {2 x^{3} y^{\prime \prime }+x^{2} \left (9+2 x y\right ) y^{\prime }+x y \left (a +3 x y-2 x^{2} y^{2}\right )=-b} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10007

ODE	\[ \boxed {2 \left (-x^{k}+4 x^{3}\right ) \left (y^{\prime \prime }+y^{\prime } y-y^{3}\right )-\left (k \,x^{k -1}-12 x^{2}\right ) \left (3 y^{\prime }+y^{2}\right )+y a x=-b} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10008

ODE	\[ \boxed {y^{\prime \prime } x^{4}+a^{2} y^{n}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10009

ODE	\[ \boxed {y^{\prime \prime } x^{4}-x \left (x^{2}+2 y\right ) y^{\prime }+4 y^{2}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (-\tanh \left (c_{1} \left (\ln \left (x \right )-c_{2} \right )\right ) c_{1} +1\right ) x^{2} \]

Problem 10010

ODE	\[ \boxed {y^{\prime \prime } x^{4}-x^{2} \left (x +y^{\prime }\right ) y^{\prime }+4 y^{2}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (-\ln \left (x \right )+c_{2} -\left (\int _{}^{\textit {\_Z}}\frac {1}{{\mathrm e}^{\textit {\_f}} c_{1} +4 \textit {\_f} +2}d \textit {\_f} \right )\right ) x^{2} \]

Problem 10011

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

Problem 10012

ODE	\[ \boxed {y^{\prime \prime } \sqrt {x}-y^{\frac {3}{2}}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10013

ODE	\[ \boxed {\left (x^{2} a +b x +c \right )^{\frac {3}{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {x^{2} a +b x +c}}\right )=0} \]
program solution
Maple solution	\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (4 \textit {\_Z} a c -\textit {\_Z} \,b^{2}-4 F \left (\frac {\textit {\_Z}}{\sqrt {a \,x^{2}+b x +c}}\right ) \sqrt {a \,x^{2}+b x +c}\right ) \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right ) a -2 a \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 c_{1} a^{2}-4 c \,\textit {\_g}^{2} a +b^{2} \textit {\_g}^{2}+8 \left (\int F \left (\textit {\_g} \right )d \textit {\_g} \right )}}d \textit {\_g} \right ) \sqrt {4 a c -b^{2}}+c_{2} \sqrt {4 a c -b^{2}}\right ) \sqrt {a \,x^{2}+b x +c} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right ) a +2 a \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 c_{1} a^{2}-4 c \,\textit {\_g}^{2} a +b^{2} \textit {\_g}^{2}+8 \left (\int F \left (\textit {\_g} \right )d \textit {\_g} \right )}}d \textit {\_g} \right ) \sqrt {4 a c -b^{2}}+c_{2} \sqrt {4 a c -b^{2}}\right ) \sqrt {a \,x^{2}+b x +c} \\ \end{align*}

Problem 10014

ODE	\[ \boxed {x^{\frac {n}{1+n}} y^{\prime \prime }-y^{\frac {1+2 n}{1+n}}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10015

ODE	\[ \boxed {f \left (x \right )^{2} y^{\prime \prime }+f \left (x \right ) f^{\prime }\left (x \right ) y^{\prime }-h \left (y, f \left (x \right ) y^{\prime }\right )=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10016

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

Problem 10017

ODE	\[ \boxed {y^{\prime \prime } y=x a} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10018

ODE	\[ \boxed {y^{\prime \prime } y=x^{2} a} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10019

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

Problem 10020

ODE	\[ \boxed {y^{\prime \prime } y+y^{2}=x a +b} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10021

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

Problem 10022

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

Problem 10023

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

Problem 10024

ODE	\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}+{\mathrm e}^{x} y \left (c y^{2}+d \right )+{\mathrm e}^{2 x} \left (b +a y^{4}\right )=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10025

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

Problem 10026

ODE	\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}-y^{\prime }+f \left (x \right ) y^{3}+y^{2} \left (\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {{f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}\right )=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10027

ODE	\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}+f \left (x \right ) y^{\prime }-y f^{\prime }\left (x \right )-y^{3}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10028

ODE	\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}+f^{\prime }\left (x \right ) y^{\prime }-f^{\prime \prime }\left (x \right ) y+f \left (x \right ) y^{3}-y^{4}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10029

ODE	\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}+a y^{\prime } y+y^{2} b=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\frac {b \textit {\_a} \left (-\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {\ln \left (\textit {\_a} \right ) a^{2}-c_{2} a^{2}-b}{b}}}{b}\right )+\frac {\ln \left (\textit {\_a} \right ) a^{2}-c_{2} a^{2}-b}{b}\right )}{a}-\textit {\_a} a \left (\ln \left (\textit {\_a} \right )-c_{2} \right )}d \textit {\_a} = c_{3} +x \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= {\mathrm e}^{\frac {{\mathrm e}^{-a x} c_{1} a +\left (-b x -c_{2} \right ) a +b}{a^{2}}} \\ \end{align*}

Problem 10030

ODE	\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}+a y^{\prime } y-2 a y^{2}+b y^{3}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10031

ODE	\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}-\left (a y-1\right ) y^{\prime }+2 a^{2} y^{2}-2 b^{2} y^{3}+a y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10032

ODE	\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}+\left (a y-1\right ) y^{\prime }-y \left (y+1\right ) \left (b^{2} y^{2}-a^{2}\right )=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10033

ODE	\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}+\left (\tan \left (x \right )+\cot \left (x \right )\right ) y y^{\prime }+\left (\cos \left (x \right )^{2}-n^{2} \cot \left (x \right )^{2}\right ) y^{2} \ln \left (y\right )=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {\pi \left (\operatorname {BesselJ}\left (n , \sin \left (x \right )\right ) c_{1} -c_{2} \operatorname {BesselY}\left (n , \sin \left (x \right )\right )\right )}{2}} \]

Problem 10034

ODE	\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}-f \left (x \right ) y^{\prime } y-g \left (x \right ) y^{2}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{c_{1} \left (\int {\mathrm e}^{\int f \left (x \right )d x}d x \right )+\int {\mathrm e}^{\int f \left (x \right )d x} \left (\int {\mathrm e}^{-\left (\int f \left (x \right )d x \right )} g \left (x \right )d x \right )d x} \]

Problem 10035

ODE	\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}+\left (g \left (x \right )+y^{2} f \left (x \right )\right ) y^{\prime }-y \left (g^{\prime }\left (x \right )-f^{\prime }\left (x \right ) y^{2}\right )=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10036

ODE	\[ \boxed {y^{\prime \prime } y-3 {y^{\prime }}^{2}+3 y^{\prime } y-y^{2}=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\frac {\textit {\_a}^{3} c_{3}^{2}}{2}+\frac {\textit {\_a}^{2} c_{3} \sqrt {2}\, \sqrt {2 \textit {\_a}^{2} c_{3}^{2}+4}}{4}+\textit {\_a}}d \textit {\_a} = x +c_{4} \] Verified OK. \[ \int _{}^{y}\frac {1}{\frac {\textit {\_a}^{3} c_{3}^{2}}{2}-\frac {\textit {\_a}^{2} c_{3} \sqrt {2}\, \sqrt {2 \textit {\_a}^{2} c_{3}^{2}+4}}{4}+\textit {\_a}}d \textit {\_a} = x +c_{5} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {{\mathrm e}^{x}}{\sqrt {2 c_{2} {\mathrm e}^{x}-2 c_{1}}} \\ y \left (x \right ) &= -\frac {{\mathrm e}^{x}}{\sqrt {2 c_{2} {\mathrm e}^{x}-2 c_{1}}} \\ \end{align*}

Problem 10037

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

Problem 10038

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

Problem 10039

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

Problem 10040

ODE	\[ \boxed {y^{\prime \prime } y+a {y^{\prime }}^{2}+b y^{\prime } y+c y^{2}+d y^{1-a}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {\left (b -\sqrt {\left (-4 a -4\right ) c +b^{2}}\right ) x}{2 a +2}} \left (\frac {\left (-4 a -4\right ) c^{3}+b^{2} c^{2}}{\left (-d \,{\mathrm e}^{-\frac {\left (-b +\sqrt {\left (-4 a -4\right ) c +b^{2}}\right ) x}{2}} \sqrt {\left (-4 a -4\right ) c +b^{2}}+\left (a +1\right ) c \left (c_{2} {\mathrm e}^{-x \sqrt {\left (-4 a -4\right ) c +b^{2}}}-c_{1} \right )\right )^{2}}\right )^{-\frac {1}{2 a +2}} \]

Problem 10041

ODE	\[ \boxed {y^{\prime \prime } y+a {y^{\prime }}^{2}+f \left (x \right ) y^{\prime } y+g \left (x \right ) y^{2}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10042

ODE	\[ \boxed {y^{\prime \prime } y+a {y^{\prime }}^{2}+b y^{2} y^{\prime }+c y^{4}=0} \]
program solution	\[ \int _{}^{y}\frac {4+2 a}{\textit {\_a}^{2} \left (\tan \left (\operatorname {RootOf}\left (-2 a \ln \left (\textit {\_a} \right ) \sqrt {4 c a -b^{2}+8 c}+2 c_{1} \sqrt {4 c a -b^{2}+8 c}\, a -\sqrt {4 c a -b^{2}+8 c}\, \ln \left (\frac {\textit {\_a}^{4} \left (4 a c \tan \left (\textit {\_Z} \right )^{2}-b^{2} \tan \left (\textit {\_Z} \right )^{2}+8 c \tan \left (\textit {\_Z} \right )^{2}+4 c a -b^{2}+8 c \right )}{4 a +8}\right )+4 c_{1} \sqrt {4 c a -b^{2}+8 c}+2 \textit {\_Z} b \right )\right ) \sqrt {4 c a -b^{2}+8 c}-b \right )}d \textit {\_a} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ \left (2 a +4\right ) \left (\int _{}^{y \left (x \right )}\frac {1}{\tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} b \,\textit {\_a}^{2}-2 a \ln \left (\textit {\_a} \right ) \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}+2 \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}\, \ln \left (2\right )-\sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}\, \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2} \left (4 a c -b^{2}+8 c \right ) \textit {\_a}^{4}}{a +2}\right )+c_{1} \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}\right )\right ) \sqrt {\textit {\_a}^{4} \left (4 \left (a +2\right ) c -b^{2}\right )}-b \,\textit {\_a}^{2}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}

Problem 10043

ODE	\[ \boxed {y^{\prime \prime } y-\frac {\left (-1+a \right ) {y^{\prime }}^{2}}{a}-y^{2} y^{\prime } f \left (x \right )+\frac {a f \left (x \right )^{2} y^{4}}{\left (a +2\right )^{2}}-\frac {a f^{\prime }\left (x \right ) y^{3}}{a +2}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10044

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

Problem 10045

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

Problem 10046

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

Problem 10047

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

Problem 10048

ODE	\[ \boxed {y^{\prime \prime } \left (x -y\right )-h \left (y^{\prime }\right )=0} \]
program solution
Maple solution	\[ y \left (x \right ) = x +\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{-1+\operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a} -1}{h \left (\textit {\_a} \right )}d \textit {\_a} +\ln \left (-\textit {\_g} \right )+c_{1} \right )}d \textit {\_g} +c_{2} \right ) \]

Problem 10049

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

Problem 10050

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

Problem 10051

ODE	\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2}+y^{2} f \left (x \right )=-a} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10052

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

Problem 10053

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

Problem 10054

ODE	\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2}-4 \left (x +2 y\right ) y^{2}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10055

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

Problem 10056

ODE	\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2}+2 y^{2} x +a y^{3}=-1} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10057

ODE	\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2}+\left (a y+b x \right ) y^{2}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10058

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

Problem 10059

ODE	\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2}-4 \left (x^{2}+a \right ) y^{2}-8 y^{3} x -3 y^{4}=-b} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10060

ODE	\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2}+3 f \left (x \right ) y^{\prime } y+2 \left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y^{2}-8 y^{3}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10061

ODE	\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2}+4 y^{\prime } y^{2}+y^{2} f \left (x \right )+y^{4}=-1} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10062

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

Problem 10063

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

Problem 10064

ODE	\[ \boxed {2 y^{\prime \prime } y-3 {y^{\prime }}^{2}+y^{2} f \left (x \right )=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10065

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

Problem 10066

ODE	\[ \boxed {2 y^{\prime \prime } y-{y^{\prime }}^{2} \left ({y^{\prime }}^{2}+1\right )=0} \]
program solution	\[ \int _{}^{y}\frac {1}{c_{2} \sqrt {\textit {\_a}}\, \sqrt {-\frac {1}{\textit {\_a} \,c_{2}^{2}-1}}}d \textit {\_a} = c_{3} +x \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right )\right ) c_{1} -2 x -2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\ y \left (x \right ) &= \frac {\left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right )\right ) c_{1} +2 x +2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\ y \left (x \right ) &= \frac {\left (-\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right )\right ) c_{1} +2 x +2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} -2 c_{2} -2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\ y \left (x \right ) &= \frac {\left (-\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right )\right ) c_{1} -2 x -2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +c_{1} \textit {\_Z} +2 c_{2} +2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\ \end{align*}

Problem 10067

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

Problem 10068

ODE	\[ \boxed {3 y^{\prime \prime } y-2 {y^{\prime }}^{2}=x^{2} a +b x +c} \]
program solution
Maple solution	\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (-2 b \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )-2 b \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 \textit {\_f}^{\frac {4}{3}} c_{1} b^{2}-36 c \,\textit {\_f}^{2} a +9 b^{2} \textit {\_f}^{2}-2}}d \textit {\_f} \right ) \sqrt {4 a c -b^{2}}+c_{2} \sqrt {4 a c -b^{2}}\right ) \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 b \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )+2 b \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 \textit {\_f}^{\frac {4}{3}} c_{1} b^{2}-36 c \,\textit {\_f}^{2} a +9 b^{2} \textit {\_f}^{2}-2}}d \textit {\_f} \right ) \sqrt {4 a c -b^{2}}+c_{2} \sqrt {4 a c -b^{2}}\right ) \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} \\ \end{align*}

Problem 10069

ODE	\[ \boxed {3 y^{\prime \prime } y-5 {y^{\prime }}^{2}=0} \]
program solution	\[ y = \frac {\left (-\frac {6}{c_{1} \left (x +c_{2} \right )}\right )^{\frac {3}{2}}}{8} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ -\frac {3}{2 y \left (x \right )^{\frac {2}{3}}}-c_{1} x -c_{2} &= 0 \\ \end{align*}

Problem 10070

ODE	\[ \boxed {4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+4 y=0} \]
program solution	\[ y = \frac {\left (c_{1}^{2} c_{2}^{2}+2 c_{1}^{2} c_{2} x +c_{1}^{2} x^{2}-16\right )^{2}}{64 c_{1}^{2}} \] Verified OK. \[ y = \frac {\left (c_{1}^{2} c_{3}^{2}+2 c_{1}^{2} c_{3} x +c_{1}^{2} x^{2}-16\right )^{2}}{64 c_{1}^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ -\frac {4 \sqrt {c_{1} y \left (x \right )^{\frac {3}{2}}+4 y \left (x \right )}}{\sqrt {y \left (x \right )}\, c_{1}}-x -c_{2} &= 0 \\ \frac {4 \sqrt {c_{1} y \left (x \right )^{\frac {3}{2}}+4 y \left (x \right )}}{\sqrt {y \left (x \right )}\, c_{1}}-x -c_{2} &= 0 \\ \end{align*}

Problem 10071

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

Problem 10072

ODE	\[ \boxed {4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+a y^{3}+y^{2} b +y c=0} \]
program solution	\[ \int _{}^{y}\frac {3}{\sqrt {-3 a \,\textit {\_a}^{3}+18 c_{1} \textit {\_a}^{\frac {3}{2}}-9 \textit {\_a}^{2} b +9 \textit {\_a} c}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {3}{\sqrt {-3 a \,\textit {\_a}^{3}+18 c_{1} \textit {\_a}^{\frac {3}{2}}-9 \textit {\_a}^{2} b +9 \textit {\_a} c}}d \textit {\_a} = c_{3} +x \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ -\sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} \left (-a \,\textit {\_a}^{2}-3 \textit {\_a} b +3 c_{1} \sqrt {\textit {\_a}}+3 c \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} \left (-a \,\textit {\_a}^{2}-3 \textit {\_a} b +3 c_{1} \sqrt {\textit {\_a}}+3 c \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}

Problem 10073

ODE	\[ \boxed {4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+\left (6 y^{2}-\frac {2 f^{\prime }\left (x \right ) y}{f \left (x \right )}\right ) y^{\prime }+y^{4}-2 y^{\prime } y^{2}+g \left (x \right ) y^{2}+f \left (x \right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10074

ODE	\[ \boxed {4 y^{\prime \prime } y-5 {y^{\prime }}^{2}+a y^{2}=0} \]
program solution	\[ -\frac {4 y \sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}\, \operatorname {arctanh}\left (\frac {\sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}}{\sqrt {a}}\right )}{\sqrt {y^{\frac {5}{2}} {\mathrm e}^{2 c_{2}} c_{3}^{2}+a y^{2}}\, \sqrt {a}} = x +c_{4} \] Verified OK. \[ \frac {4 y \sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}\, \operatorname {arctanh}\left (\frac {\sqrt {{\mathrm e}^{2 c_{2}} c_{3}^{2} \sqrt {y}+a}}{\sqrt {a}}\right )}{\sqrt {y^{\frac {5}{2}} {\mathrm e}^{2 c_{2}} c_{3}^{2}+a y^{2}}\, \sqrt {a}} = x +c_{5} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {16 \,{\mathrm e}^{\sqrt {a}\, x} a^{2}}{\left ({\mathrm e}^{\frac {\sqrt {a}\, x}{2}} c_{1} -c_{2} \right )^{4}} \\ \end{align*}

Problem 10075

ODE	\[ \boxed {12 y^{\prime \prime } y-15 {y^{\prime }}^{2}+8 y^{3}=0} \]
program solution	\[ y = \frac {576 c_{1}^{2}}{\left (3 c_{1}^{2} c_{2}^{2}+6 c_{1}^{2} c_{2} x +3 c_{1}^{2} x^{2}+32\right )^{2}} \] Verified OK. \[ y = \frac {576 c_{1}^{2}}{\left (3 c_{1}^{2} c_{3}^{2}+6 c_{1}^{2} c_{3} x +3 c_{1}^{2} x^{2}+32\right )^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ -\frac {12 y \left (x \right ) \left (8 \sqrt {y \left (x \right )}-c_{1} \right ) \sqrt {8 y \left (x \right )-c_{1} \sqrt {y \left (x \right )}}}{\sqrt {-24 y \left (x \right )^{3}+3 c_{1} y \left (x \right )^{\frac {5}{2}}}\, c_{1} \sqrt {\sqrt {y \left (x \right )}\, \left (8 \sqrt {y \left (x \right )}-c_{1} \right )}}-x -c_{2} &= 0 \\ \frac {12 y \left (x \right ) \left (8 \sqrt {y \left (x \right )}-c_{1} \right ) \sqrt {8 y \left (x \right )-c_{1} \sqrt {y \left (x \right )}}}{\sqrt {-24 y \left (x \right )^{3}+3 c_{1} y \left (x \right )^{\frac {5}{2}}}\, c_{1} \sqrt {\sqrt {y \left (x \right )}\, \left (8 \sqrt {y \left (x \right )}-c_{1} \right )}}-x -c_{2} &= 0 \\ \end{align*}

Problem 10076

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

Problem 10077

ODE	\[ \boxed {a y y^{\prime \prime }+{y^{\prime }}^{2} b +\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y=-\operatorname {c0}} \]
program solution	\[ \int _{}^{y}\frac {\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right )}{\sqrt {-\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right ) \left (12 a \operatorname {c4} \,b^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}-8 c_{1} b^{5}+12 b \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{3}+3 b \operatorname {c4} \,a^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}+11 \operatorname {c4} \,a^{2} b^{2} \textit {\_a}^{\frac {4 a +2 b}{a}}-12 c_{1} a^{4} b -50 c_{1} a^{3} b^{2}-70 c_{1} a^{2} b^{3}-40 c_{1} a \,b^{4}+6 b \operatorname {c2} \,a^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+20 \textit {\_a}^{\frac {2 b}{a}} a \,b^{3} \operatorname {c0} +35 \textit {\_a}^{\frac {2 b}{a}} a^{2} b^{2} \operatorname {c0} +4 \operatorname {c4} \,b^{4} \textit {\_a}^{\frac {4 a +2 b}{a}}+4 \operatorname {c2} \,b^{4} \textit {\_a}^{\frac {2 a +2 b}{a}}+4 \operatorname {c3} \,b^{4} \textit {\_a}^{\frac {3 a +2 b}{a}}+6 \textit {\_a}^{\frac {2 b}{a}} a^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {2 b}{a}} b^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{4}+18 a \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{3}+4 b \operatorname {c3} \,a^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+26 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{2} b^{2}+19 \operatorname {c2} \,a^{2} b^{2} \textit {\_a}^{\frac {2 a +2 b}{a}}+16 a \operatorname {c2} \,b^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+14 \operatorname {c3} \,a^{2} b^{2} \textit {\_a}^{\frac {3 a +2 b}{a}}+25 \textit {\_a}^{\frac {2 b}{a}} a^{3} b \operatorname {c0} +14 a \operatorname {c3} \,b^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}\right )}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right )}{\sqrt {-\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 a^{3} b +35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right ) \left (-8 c_{1} b^{5}-12 c_{1} a^{4} b -50 c_{1} a^{3} b^{2}-70 c_{1} a^{2} b^{3}-40 c_{1} a \,b^{4}+4 \operatorname {c3} \,b^{4} \textit {\_a}^{\frac {3 a +2 b}{a}}+6 \textit {\_a}^{\frac {2 b}{a}} a^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {2 b}{a}} b^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{4}+4 \operatorname {c2} \,b^{4} \textit {\_a}^{\frac {2 a +2 b}{a}}+4 \operatorname {c4} \,b^{4} \textit {\_a}^{\frac {4 a +2 b}{a}}+6 b \operatorname {c2} \,a^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+19 \operatorname {c2} \,a^{2} b^{2} \textit {\_a}^{\frac {2 a +2 b}{a}}+16 a \operatorname {c2} \,b^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+3 b \operatorname {c4} \,a^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}+11 \operatorname {c4} \,a^{2} b^{2} \textit {\_a}^{\frac {4 a +2 b}{a}}+4 b \operatorname {c3} \,a^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+14 \operatorname {c3} \,a^{2} b^{2} \textit {\_a}^{\frac {3 a +2 b}{a}}+25 \textit {\_a}^{\frac {2 b}{a}} a^{3} b \operatorname {c0} +35 \textit {\_a}^{\frac {2 b}{a}} a^{2} b^{2} \operatorname {c0} +20 \textit {\_a}^{\frac {2 b}{a}} a \,b^{3} \operatorname {c0} +12 b \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{3}+26 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{2} b^{2}+18 a \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{3}+14 a \operatorname {c3} \,b^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+12 a \operatorname {c4} \,b^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}\right )}}d \textit {\_a} = c_{3} +x \] Verified OK.
Maple solution	\begin{align*} 6 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \left (a +\frac {2 b}{3}\right ) \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{\frac {2 b}{a}}}{\sqrt {-36 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \,\textit {\_a}^{\frac {2 b}{a}} \left (\frac {2 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \operatorname {c3} \,\textit {\_a}^{\frac {3 a +2 b}{a}}}{3}+\left (\left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \operatorname {c2} \,\textit {\_a}^{\frac {2 a +2 b}{a}}+\left (a +b \right ) \left (\frac {\operatorname {c4} b \left (a +2 b \right ) \textit {\_a}^{\frac {4 a +2 b}{a}}}{2}+\left (2 \textit {\_a}^{\frac {a +2 b}{a}} b \operatorname {c1} +\left (\textit {\_a}^{\frac {2 b}{a}} \operatorname {c0} -c_{1} b \right ) \left (a +2 b \right )\right ) \left (a +\frac {b}{2}\right )\right )\right ) \left (a +\frac {2 b}{3}\right )\right ) \left (a +2 b \right ) \left (a +\frac {2 b}{3}\right )}}d \textit {\_a} \right )-c_{2} -x &= 0 \\ -6 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \left (a +\frac {2 b}{3}\right ) \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{\frac {2 b}{a}}}{\sqrt {-36 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \,\textit {\_a}^{\frac {2 b}{a}} \left (\frac {2 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \operatorname {c3} \,\textit {\_a}^{\frac {3 a +2 b}{a}}}{3}+\left (\left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \operatorname {c2} \,\textit {\_a}^{\frac {2 a +2 b}{a}}+\left (a +b \right ) \left (\frac {\operatorname {c4} b \left (a +2 b \right ) \textit {\_a}^{\frac {4 a +2 b}{a}}}{2}+\left (2 \textit {\_a}^{\frac {a +2 b}{a}} b \operatorname {c1} +\left (\textit {\_a}^{\frac {2 b}{a}} \operatorname {c0} -c_{1} b \right ) \left (a +2 b \right )\right ) \left (a +\frac {b}{2}\right )\right )\right ) \left (a +\frac {2 b}{3}\right )\right ) \left (a +2 b \right ) \left (a +\frac {2 b}{3}\right )}}d \textit {\_a} \right )-c_{2} -x &= 0 \\ \end{align*}

Problem 10078

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

Problem 10079

ODE	\[ \boxed {a y y^{\prime \prime }-\left (-1+a \right ) {y^{\prime }}^{2}+\left (a +2\right ) f \left (x \right ) y^{2} y^{\prime }+f \left (x \right )^{2} y^{4}+a f^{\prime }\left (x \right ) y^{3}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10080

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

Problem 10081

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

Problem 10082

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

Problem 10083

ODE	\[ \boxed {x y y^{\prime \prime }-x {y^{\prime }}^{2}+y^{\prime } y+x \left (d +a y^{4}\right )+y \left (c +y^{2} b \right )=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10084

ODE	\[ \boxed {x y y^{\prime \prime }-x {y^{\prime }}^{2}+a y^{\prime } y+b x y^{3}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10085

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

Problem 10086

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

Problem 10087

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

Problem 10088

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

Problem 10089

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

Problem 10090

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

Problem 10091

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

Problem 10092

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

Problem 10093

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

Problem 10094

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

Problem 10095

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

Problem 10096

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

Problem 10097

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

Problem 10098

ODE	\[ \boxed {8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 y x^{2} y^{\prime }+3 y^{2} x=0} \]
program solution
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {{\left (\operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) c_{1} +\frac {c_{2} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{2}\right )}^{2} x}{c_{1}} \\ \end{align*}

Problem 10099

ODE	\[ \boxed {\operatorname {f0} \left (x \right ) y y^{\prime \prime }+\operatorname {f1} \left (x \right ) {y^{\prime }}^{2}+\operatorname {f2} \left (x \right ) y y^{\prime }+\operatorname {f3} \left (x \right ) y^{2}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10100

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