Internal problem ID [9407]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1828.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [NONE]
Solve \begin {gather*} \boxed {2 \left (x^{2}+1\right ) \left (y^{\prime \prime }\right )^{2}-x y^{\prime \prime } \left (x +4 y^{\prime }\right )+2 \left (x +y^{\prime }\right ) y^{\prime }-2 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.179 (sec). Leaf size: 67
dsolve(2*(x^2+1)*diff(diff(y(x),x),x)^2-x*diff(diff(y(x),x),x)*(x+4*diff(y(x),x))+2*(x+diff(y(x),x))*diff(y(x),x)-2*y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = c_{1} \left (\frac {x \sqrt {x^{2}+1}}{2}+\frac {\arcsinh \relax (x )}{2}\right )-\frac {3 x^{2}}{16}+\frac {\arcsinh \relax (x ) \sqrt {x^{2}+1}\, x}{8}+\frac {\arcsinh \relax (x )^{2}}{16}+c_{1}^{2} \\ y \relax (x ) = \frac {1}{2} c_{1} x^{2}+x c_{2}+c_{1}^{2}+c_{2}^{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 32
DSolve[-2*y[x] + 2*y'[x]*(x + y'[x]) - x*(x + 4*y'[x])*y''[x] + 2*(1 + x^2)*y''[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} \sqrt {c_2-c_1{}^2} x^2+c_1 x+c_2 \\ \end{align*}