53.1.13 problem 13

Internal problem ID [8447]
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
Date solved : Monday, January 27, 2025 at 04:01:19 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} \left (x -y\right )^{2} {y^{\prime }}^{2}&=y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 47

dsolve((x-y(x))^2*diff(y(x),x)^2=y(x)^2,y(x), singsol=all)
 

\begin{align*} y &= x -\sqrt {x^{2}-2 c_{1}} \\ y &= x +\sqrt {x^{2}-2 c_{1}} \\ y &= -\frac {x}{\operatorname {LambertW}\left (-x \,{\mathrm e}^{-c_{1}}\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 4.439 (sec). Leaf size: 99

DSolve[(x-y[x])^2*(D[y[x],x])^2==y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to x-\sqrt {x^2-e^{2 c_1}} \\ y(x)\to x+\sqrt {x^2-e^{2 c_1}} \\ y(x)\to -\frac {x}{W\left (-e^{-c_1} x\right )} \\ y(x)\to 0 \\ y(x)\to x-\sqrt {x^2} \\ y(x)\to \sqrt {x^2}+x \\ \end{align*}