Internal problem ID [8024]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 444.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {x^{2} \left (y^{\prime }\right )^{2}-y \left (y-2 x \right ) y^{\prime }+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.322 (sec). Leaf size: 125
dsolve(x^2*diff(y(x),x)^2-y(x)*(y(x)-2*x)*diff(y(x),x)+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 4 x \\ y \relax (x ) = 0 \\ y \relax (x ) = -\frac {2 c_{1}^{2} \left (\sqrt {2}\, c_{1}-x \right )}{2 c_{1}^{2}-x^{2}} \\ y \relax (x ) = \frac {2 c_{1}^{2} \left (\sqrt {2}\, c_{1}+x \right )}{2 c_{1}^{2}-x^{2}} \\ y \relax (x ) = -\frac {c_{1}^{2} \left (\sqrt {2}\, c_{1}-2 x \right )}{2 \left (c_{1}^{2}-2 x^{2}\right )} \\ y \relax (x ) = \frac {c_{1}^{2} \left (\sqrt {2}\, c_{1}+2 x \right )}{2 c_{1}^{2}-4 x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.692 (sec). Leaf size: 61
DSolve[y[x]^2 - y[x]*(-2*x + y[x])*y'[x] + x^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{-4 e^{4 c_1} x-2 e^{2 c_1}} \\ y(x)\to \frac {1}{2 e^{2 c_1}-4 e^{4 c_1} x} \\ y(x)\to 0 \\ y(x)\to 4 x \\ \end{align*}