Internal problem ID [5022]
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: 27.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {y \left (y^{\prime }\right )^{2}+2 x y^{\prime }-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.344 (sec). Leaf size: 75
dsolve(y(x)*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x \\ y \relax (x ) = i x \\ y \relax (x ) = 0 \\ y \relax (x ) = \sqrt {-2 c_{1} x +c_{1}^{2}} \\ y \relax (x ) = \sqrt {2 c_{1} x +c_{1}^{2}} \\ y \relax (x ) = -\sqrt {-2 c_{1} x +c_{1}^{2}} \\ y \relax (x ) = -\sqrt {2 c_{1} x +c_{1}^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.714 (sec). Leaf size: 126
DSolve[y[x]*(y'[x])^2+2*x*y'[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}} \\ y(x)\to e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}} \\ y(x)\to -e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}} \\ y(x)\to e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to -i x \\ y(x)\to i x \\ \end{align*}