Internal problem ID [5772]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.2 Homogeneous equations
problems. page 12
Problem number: 24.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _dAlembert]
\[ \boxed {\left (y^{\prime } x +y\right )^{2}-y^{2} y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.235 (sec). Leaf size: 125
dsolve((x*diff(y(x),x)+y(x))^2=y(x)^2*diff(y(x),x),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 (\sqrt {2}\, c_{1} -x \right )}{2 c_{1}^{2}-x^{2}} y \left (x \right ) = \frac {2 c_{1}^{2} \left (\sqrt {2}\, c_{1} +x \right )}{2 c_{1}^{2}-x^{2}} y \left (x \right ) = -\frac {c_{1}^{2} \left (\sqrt {2}\, c_{1} -2 x \right )}{2 \left (c_{1}^{2}-2 x^{2}\right )} y \left (x \right ) = \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: 0.501 (sec). Leaf size: 62
DSolve[(x*y'[x]+y[x])^2==y[x]^2*y'[x],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*}