23.1.583 problem 576

Internal problem ID [5190]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 576
Date solved : Tuesday, September 30, 2025 at 11:51:55 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.464 (sec). Leaf size: 57

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

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

✓ Sympy. Time used: 0.368 (sec). Leaf size: 29

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

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