53.1.9 problem 9

Internal problem ID [8443]
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 : 9
Date solved : Monday, January 27, 2025 at 04:00:57 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: 0.030 (sec). Leaf size: 47

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

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

✓ Solution by Mathematica

Time used: 4.168 (sec). Leaf size: 101

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 -\sqrt {x^2}-x \\ y(x)\to \sqrt {x^2}-x \\ \end{align*}