89.4.19 problem 20

Internal problem ID [24341]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 39
Problem number : 20
Date solved : Thursday, October 02, 2025 at 10:20:35 PM
CAS classification : [_exact, _rational]

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

With initial conditions

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 49

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

\begin{align*} y &= -\frac {\sqrt {9 x^{2}-6 \sqrt {3 x^{3}+12}}}{3} \\ y &= \frac {\sqrt {9 x^{2}-6 \sqrt {3 x^{3}+12}}}{3} \\ \end{align*}

✓ Mathematica. Time used: 2.525 (sec). Leaf size: 59

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

\begin{align*} y(x)&\to -\sqrt {x^2-\frac {2 \sqrt {x^3+4}}{\sqrt {3}}}\\ y(x)&\to \sqrt {x^2-\frac {2 \sqrt {x^3+4}}{\sqrt {3}}} \end{align*}

✗ Sympy

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

Timed Out