Internal problem ID [9408]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1830 (book 6.239).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {3 \left (y^{\prime \prime }\right )^{2} x^{2}-2 \left (3 y^{\prime } x +y\right ) y^{\prime \prime }+4 \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.266 (sec). Leaf size: 36
dsolve(3*x^2*diff(diff(y(x),x),x)^2-2*(3*x*diff(y(x),x)+y(x))*diff(diff(y(x),x),x)+4*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = x^{\frac {2 \sqrt {3}}{3}} c_{1} x \\ y \relax (x ) = 0 \\ y \relax (x ) = \frac {c_{1}^{2} x^{2}}{c_{2}}+c_{1} x +c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 24
DSolve[4*y'[x]^2 - 2*(y[x] + 3*x*y'[x])*y''[x] + 3*x^2*y''[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1{}^2 x^2}{c_2}+c_1 x+c_2 \\ \end{align*}