Internal problem ID [9268]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1690 (book 6.99).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{4} y^{\prime \prime }+\left (y^{\prime } x -y\right )^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 37
dsolve(x^4*diff(diff(y(x),x),x)+(x*diff(y(x),x)-y(x))^3=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \left (-\arctan \left (\frac {1}{\sqrt {c_{1} x^{2}-1}}\right )+c_{2}\right ) x \\ y \relax (x ) = \left (\arctan \left (\frac {1}{\sqrt {c_{1} x^{2}-1}}\right )+c_{2}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.198 (sec). Leaf size: 95
DSolve[(-y[x] + x*y'[x])^3 + x^4*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -i x \log \left (\frac {e^{c_2}-\sqrt {e^{2 c_2}-8 i c_1 x^2}}{4 c_1 x}\right ) \\ y(x)\to -i x \log \left (\frac {\sqrt {e^{2 c_2}-8 i c_1 x^2}+e^{c_2}}{4 c_1 x}\right ) \\ \end{align*}