23.1.676 problem 671

Internal problem ID [5283]
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 : 671
Date solved : Tuesday, September 30, 2025 at 12:06:56 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 61

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

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

✓ Mathematica. Time used: 0.283 (sec). Leaf size: 62

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]^2+1}{(K[1]-1) K[1] (K[1]+1)}dK[1]\&\right ]\left [\log \left (x^2+1\right )+c_1\right ]\\ y(x)&\to -1\\ y(x)&\to 0\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 7.368 (sec). Leaf size: 71

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

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