Internal problem ID [9352]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1774 (book 6.183).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 y^{\prime \prime } y x^{2}-x^{2} \left (\left (y^{\prime }\right )^{2}+1\right )+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 28
dsolve(2*x^2*y(x)*diff(diff(y(x),x),x)-x^2*(diff(y(x),x)^2+1)+y(x)^2=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (c_{1}^{2}+1\right ) x}{4 c_{2}}+x \ln \relax (x ) c_{1}+c_{2} x \ln \relax (x )^{2} \]
✓ Solution by Mathematica
Time used: 0.261 (sec). Leaf size: 39
DSolve[y[x]^2 - x^2*(1 + y'[x]^2) + 2*x^2*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \left (c_1{}^2 \log (x) (\log (x)-2 c_2)+4+c_2{}^2 c_1{}^2\right )}{4 c_1} \\ \end{align*}