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]
\[ \boxed {\left (x y^{\prime }+y\right )^{2}-y^{\prime } y^{2}=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 124
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}^{3} \sqrt {2}-2 c_{1}^{2} x}{-2 c_{1}^{2}+4 x^{2}} \\ 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*}