89.32.18 problem 21

Internal problem ID [24978]
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. Miscellaneous Exercises at page 246
Problem number : 21
Date solved : Thursday, October 02, 2025 at 11:45:16 PM
CAS classification : [_linear]

\begin{align*} {y^{\prime }}^{2} x^{2}-\left (x -y\right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

ode:=x^2*diff(y(x),x)^2-(x-y(x))^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \left (-\ln \left (x \right )+c_1 \right ) x \\ y &= \frac {x}{2}+\frac {c_1}{x} \\ \end{align*}

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 30

ode=x^2*D[y[x],x]^2-(x-y[x])^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x}{2}+\frac {c_1}{x}\\ y(x)&\to x (-\log (x)+c_1) \end{align*}

✓ Sympy. Time used: 0.170 (sec). Leaf size: 17

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

\[ \left [ y{\left (x \right )} = x \left (C_{1} - \log {\left (x \right )}\right ), \ y{\left (x \right )} = \frac {C_{1}}{x} + \frac {x}{2}\right ] \]