Internal problem ID [9182]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1603 (6.13).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [NONE]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {1}{\left (a y^{2}+b x y+c \,x^{2}+\alpha y+\beta x +\gamma \right )^{\frac {3}{2}}}=0} \end {gather*}
✓ Solution by Maple
Time used: 1.515 (sec). Leaf size: 941
dsolve(diff(y(x),x$2)-(a*y(x)^2+b*x*y(x)+c*x^2+alpha*y(x)+beta*x+gamma)^(-3/2)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {2 \RootOf \left (-2 \arctan \left (\frac {4 c x a -b^{2} x +2 a \beta -\alpha b}{2 \sqrt {-a \left (a \,\beta ^{2}-4 a c \gamma +\alpha ^{2} c -\alpha \beta b +b^{2} \gamma \right )}}\right ) a \beta +\arctan \left (\frac {4 c x a -b^{2} x +2 a \beta -\alpha b}{2 \sqrt {-a \left (a \,\beta ^{2}-4 a c \gamma +\alpha ^{2} c -\alpha \beta b +b^{2} \gamma \right )}}\right ) \alpha b +2 \left (\int _{}^{\textit {\_Z}}\frac {2 a \beta -\alpha b}{\sqrt {16 \sqrt {4 a \beta +4 c a +4 a \gamma -\alpha ^{2}-2 \alpha b -b^{2}}\, \left (\int \frac {1}{\left (4 \textit {\_g}^{2} a^{2}+1\right ) \sqrt {\frac {16 a^{3} \textit {\_g}^{2} \beta +16 c \,\textit {\_g}^{2} a^{3}+16 a^{3} \textit {\_g}^{2} \gamma -4 a^{2} \textit {\_g}^{2} \alpha ^{2}-8 \textit {\_g}^{2} \alpha b \,a^{2}-4 \textit {\_g}^{2} b^{2} a^{2}+4 a \beta +4 c a +4 a \gamma -\alpha ^{2}-2 \alpha b -b^{2}}{a}}}d \textit {\_g} \right ) a +4 a^{2} \beta ^{2} \textit {\_g}^{2}-16 a^{2} c \gamma \,\textit {\_g}^{2}+4 c_{1} a^{2} \beta ^{2}+4 a \,\alpha ^{2} c \,\textit {\_g}^{2}-4 a \alpha b \beta \,\textit {\_g}^{2}+4 a \,b^{2} \gamma \,\textit {\_g}^{2}-4 c_{1} a \alpha b \beta +c_{1} \alpha ^{2} b^{2}}}d \textit {\_g} \right ) \sqrt {-a \left (a \,\beta ^{2}-4 a c \gamma +\alpha ^{2} c -\alpha \beta b +b^{2} \gamma \right )}+2 c_{2} \sqrt {-a \left (a \,\beta ^{2}-4 a c \gamma +\alpha ^{2} c -\alpha \beta b +b^{2} \gamma \right )}\right ) \sqrt {4 a c \,x^{2}-b^{2} x^{2}+4 a \beta x -2 \alpha b x +4 a \gamma -\alpha ^{2}}\, a -b x -\alpha }{2 a} \\ y \relax (x ) = \frac {2 \RootOf \left (-2 \arctan \left (\frac {4 c x a -b^{2} x +2 a \beta -\alpha b}{2 \sqrt {-a \left (a \,\beta ^{2}-4 a c \gamma +\alpha ^{2} c -\alpha \beta b +b^{2} \gamma \right )}}\right ) a \beta +\arctan \left (\frac {4 c x a -b^{2} x +2 a \beta -\alpha b}{2 \sqrt {-a \left (a \,\beta ^{2}-4 a c \gamma +\alpha ^{2} c -\alpha \beta b +b^{2} \gamma \right )}}\right ) \alpha b -2 \left (\int _{}^{\textit {\_Z}}\frac {2 a \beta -\alpha b}{\sqrt {16 \sqrt {4 a \beta +4 c a +4 a \gamma -\alpha ^{2}-2 \alpha b -b^{2}}\, \left (\int \frac {1}{\left (4 \textit {\_g}^{2} a^{2}+1\right ) \sqrt {\frac {16 a^{3} \textit {\_g}^{2} \beta +16 c \,\textit {\_g}^{2} a^{3}+16 a^{3} \textit {\_g}^{2} \gamma -4 a^{2} \textit {\_g}^{2} \alpha ^{2}-8 \textit {\_g}^{2} \alpha b \,a^{2}-4 \textit {\_g}^{2} b^{2} a^{2}+4 a \beta +4 c a +4 a \gamma -\alpha ^{2}-2 \alpha b -b^{2}}{a}}}d \textit {\_g} \right ) a +4 a^{2} \beta ^{2} \textit {\_g}^{2}-16 a^{2} c \gamma \,\textit {\_g}^{2}+4 c_{1} a^{2} \beta ^{2}+4 a \,\alpha ^{2} c \,\textit {\_g}^{2}-4 a \alpha b \beta \,\textit {\_g}^{2}+4 a \,b^{2} \gamma \,\textit {\_g}^{2}-4 c_{1} a \alpha b \beta +c_{1} \alpha ^{2} b^{2}}}d \textit {\_g} \right ) \sqrt {-a \left (a \,\beta ^{2}-4 a c \gamma +\alpha ^{2} c -\alpha \beta b +b^{2} \gamma \right )}+2 c_{2} \sqrt {-a \left (a \,\beta ^{2}-4 a c \gamma +\alpha ^{2} c -\alpha \beta b +b^{2} \gamma \right )}\right ) \sqrt {4 a c \,x^{2}-b^{2} x^{2}+4 a \beta x -2 \alpha b x +4 a \gamma -\alpha ^{2}}\, a -b x -\alpha }{2 a} \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]-(a*y[x]^2+b*x*y[x]+c*x^2+\[Alpha]*y[x]+\[Beta]*x+\[Gamma])^(-3/2) == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved