89.28.13 problem 13

Internal problem ID [24901]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Exercises at page 229
Problem number : 13
Date solved : Thursday, October 02, 2025 at 10:49:16 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*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 47

ode:=(x-y(x))^2*diff(y(x),x)^2 = y(x)^2; 
dsolve(ode,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*}

✓ Mathematica. Time used: 3.04 (sec). Leaf size: 99

ode=(x-y[x])^2*D[y[x],x]^2==y[x]^2; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 1.382 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - y(x))**2*Derivative(y(x), x)**2 - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = e^{C_{1} + W\left (- x e^{- C_{1}}\right )}, \ y{\left (x \right )} = x - \sqrt {C_{1} + x^{2}}, \ y{\left (x \right )} = x + \sqrt {C_{1} + x^{2}}\right ] \]