Internal problem ID [8021]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 441.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {x^{2} \left (y^{\prime }\right )^{2}-4 x \left (y+2\right ) y^{\prime }+4 y \left (y+2\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 1.641 (sec). Leaf size: 118
dsolve(x^2*diff(y(x),x)^2-4*x*(y(x)+2)*diff(y(x),x)+4*y(x)*(y(x)+2) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -2 \\ y \relax (x ) = -\frac {\left (-\frac {2 \sqrt {2}\, \sqrt {x^{2} c_{1}}}{x^{2}}-1\right ) x^{2}}{c_{1}} \\ y \relax (x ) = -\frac {\left (\frac {2 \sqrt {2}\, \sqrt {x^{2} c_{1}}}{x^{2}}-1\right ) x^{2}}{c_{1}} \\ y \relax (x ) = \frac {2 c_{1} \left (-\sqrt {2}\, x +4 c_{1}\right )-8 c_{1}^{2}+x^{2}}{c_{1}^{2}} \\ y \relax (x ) = \frac {2 c_{1} \left (\sqrt {2}\, x +4 c_{1}\right )-8 c_{1}^{2}+x^{2}}{c_{1}^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.193 (sec). Leaf size: 69
DSolve[4*y[x]*(2 + y[x]) - 4*x*(2 + y[x])*y'[x] + x^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-c_1} x \left (x-2 \sqrt {2} e^{\frac {c_1}{2}}\right ) \\ y(x)\to e^{c_1} x^2-2 \sqrt {2} e^{\frac {c_1}{2}} x \\ y(x)\to -2 \\ y(x)\to 0 \\ \end{align*}