Internal problem ID [5019]
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]
Solve \begin {gather*} \boxed {\left (x y^{\prime }+y\right )^{2}-y^{2} y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.329 (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 \relax (x ) = 4 x \\ y \relax (x ) = 0 \\ y \relax (x ) = -\frac {2 c_{1}^{2} \left (\sqrt {2}\, c_{1}-x \right )}{-x^{2}+2 c_{1}^{2}} \\ y \relax (x ) = \frac {2 c_{1}^{2} \left (\sqrt {2}\, c_{1}+x \right )}{-x^{2}+2 c_{1}^{2}} \\ y \relax (x ) = -\frac {c_{1}^{2} \left (\sqrt {2}\, c_{1}-2 x \right )}{2 \left (-2 x^{2}+c_{1}^{2}\right )} \\ y \relax (x ) = \frac {c_{1}^{2} \left (\sqrt {2}\, c_{1}+2 x \right )}{-4 x^{2}+2 c_{1}^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.997 (sec). Leaf size: 61
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 {1}{-4 e^{4 c_1} x-2 e^{2 c_1}} \\ y(x)\to 0 \\ y(x)\to 4 x \\ \end{align*}