Internal problem ID [6026]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 94. Factoring the left member. EXERCISES
Page 309
Problem number: 13.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {\left (x -y\right )^{2} \left (y^{\prime }\right )^{2}-y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 45
dsolve((x-y(x))^2*diff(y(x),x)^2=y(x)^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = x -\sqrt {x^{2}-2 c_{1}} \\ y \relax (x ) = x +\sqrt {x^{2}-2 c_{1}} \\ y \relax (x ) = {\mathrm e}^{\LambertW \left (-x \,{\mathrm e}^{-c_{1}}\right )+c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.549 (sec). Leaf size: 93
DSolve[(x-y[x])^2*(y'[x])^2==y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x-\sqrt {x^2-c_1{}^2} \\ y(x)\to x+\sqrt {x^2-c_1{}^2} \\ y(x)\to c_1 e^{\text {ProductLog}\left (-\frac {x}{c_1}\right )} \\ y(x)\to 0 \\ y(x)\to x-\sqrt {x^2} \\ y(x)\to \sqrt {x^2}+x \\ \end{align*}