Internal problem ID [9345]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1767 (book 6.176).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x y y^{\prime \prime }-4 x \left (y^{\prime }\right )^{2}+4 y^{\prime } y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 88
dsolve(x*y(x)*diff(diff(y(x),x),x)-4*x*diff(y(x),x)^2+4*y(x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \frac {x}{\left (-3 x^{3} c_{2}+c_{1}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \left (-\frac {1}{2 \left (-3 x^{3} c_{2}+c_{1}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}}{2 \left (-3 x^{3} c_{2}+c_{1}\right )^{\frac {1}{3}}}\right ) x \\ y \relax (x ) = \left (-\frac {1}{2 \left (-3 x^{3} c_{2}+c_{1}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}}{2 \left (-3 x^{3} c_{2}+c_{1}\right )^{\frac {1}{3}}}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.108 (sec). Leaf size: 21
DSolve[4*y[x]*y'[x] - 4*x*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 x}{\sqrt [3]{1+c_1 x^3}} \\ \end{align*}