Internal problem ID [9262]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1683.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime }-a \left (y^{\prime } x -y\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.023 (sec). Leaf size: 23
dsolve(x^3*diff(diff(y(x),x),x)-a*(x*diff(y(x),x)-y(x))^2=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\ln \left (\frac {a \left (x c_{1}-c_{2}\right )}{x}\right ) x}{a} \]
✓ Solution by Mathematica
Time used: 0.057 (sec). Leaf size: 25
DSolve[-(a*(-y[x] + x*y'[x])^2) + x^3*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x \log \left (-\frac {a (c_2 x+c_1)}{x}\right )}{a} \\ \end{align*}