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 : Sunday, March 30, 2025 at 01:05:09 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: 0.024 (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: 4.439 (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: 2.093 (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)
 

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