89.4.8 problem 9

Internal problem ID [24330]
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 : 9
Date solved : Thursday, October 02, 2025 at 10:18:22 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} \left (1+x^{2}-y^{2}\right ) y-x \left (-y^{2}+x^{2}-1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 159

ode:=y(x)*(x^2-y(x)^2+1)-x*(x^2-y(x)^2-1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.717 (sec). Leaf size: 106

ode=y[x]*( x^2-y[x]^2+1)-x*( x^2-y[x]^2-1)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {-\sqrt {x^2-4 c_1 x^2+4 c_1{}^2}+x-2 c_1 x}{2 c_1}\\ y(x)&\to \frac {\sqrt {x^2-4 c_1 x^2+4 c_1{}^2}+x-2 c_1 x}{2 c_1}\\ y(x)&\to \text {Indeterminate}\\ y(x)&\to -x-1\\ y(x)&\to 1-x \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(x**2 - y(x)**2 - 1)*Derivative(y(x), x) + (x**2 - y(x)**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out