89.3.18 problem 18

Internal problem ID [24316]
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 34
Problem number : 18
Date solved : Thursday, October 02, 2025 at 10:17:42 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.057 (sec). Leaf size: 47

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

\begin{align*} y &= \frac {1-\sqrt {-4 c_1^{2} x^{2}+1}}{2 c_1} \\ y &= \frac {1+\sqrt {-4 c_1^{2} x^{2}+1}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.586 (sec). Leaf size: 66

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

\begin{align*} y(x)&\to \frac {1}{2} \left (e^{c_1}-\sqrt {-4 x^2+e^{2 c_1}}\right )\\ y(x)&\to \frac {1}{2} \left (\sqrt {-4 x^2+e^{2 c_1}}+e^{c_1}\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.990 (sec). Leaf size: 44

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {- 4 x^{2} + e^{2 C_{1}}}}{2} + \frac {e^{C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {- 4 x^{2} + e^{2 C_{1}}}}{2} + \frac {e^{C_{1}}}{2}\right ] \]