Internal problem ID [8779]
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]
\[ \boxed {x^{2} {y^{\prime }}^{2}-y \left (y-2 x \right ) y^{\prime }+y^{2}=0} \]
✓ Solution by Maple
Time used: 0.078 (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 \left (x \right ) = 4 x y \left (x \right ) = 0 y \left (x \right ) = -\frac {2 c_{1}^{2} \left (c_{1} \sqrt {2}-x \right )}{2 c_{1}^{2}-x^{2}} y \left (x \right ) = \frac {2 c_{1}^{2} \left (c_{1} \sqrt {2}+x \right )}{2 c_{1}^{2}-x^{2}} y \left (x \right ) = -\frac {c_{1}^{2} \left (c_{1} \sqrt {2}-2 x \right )}{2 \left (c_{1}^{2}-2 x^{2}\right )} y \left (x \right ) = \frac {c_{1}^{2} \left (c_{1} \sqrt {2}+2 x \right )}{2 c_{1}^{2}-4 x^{2}} \end{align*}
✓ Solution by Mathematica
Time used: 0.639 (sec). Leaf size: 62
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 {4 e^{-2 c_1}}{2+e^{2 c_1} x} y(x)\to -\frac {e^{-2 c_1}}{2+4 e^{2 c_1} x} y(x)\to 0 y(x)\to 4 x \end{align*}