23.1.698 problem 694

Internal problem ID [5305]
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 : 694
Date solved : Tuesday, September 30, 2025 at 12:28:49 PM
CAS classification : [_separable]

\begin{align*} x y^{3} y^{\prime }&=\left (-x^{2}+1\right ) \left (1+y^{2}\right ) \end{align*}

✓ Maple. Time used: 0.135 (sec). Leaf size: 29

ode:=x*y(x)^3*diff(y(x),x) = (-x^2+1)*(1+y(x)^2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {x^{2}}{2}-\ln \left (x \right )+\frac {y^{2}}{2}-\frac {\ln \left (1+y^{2}\right )}{2}+c_1 = 0 \]

✓ Mathematica. Time used: 0.185 (sec). Leaf size: 54

ode=x y[x]^3 D[y[x],x]==(1-x^2)(1+y[x]^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]^3}{K[1]^2+1}dK[1]\&\right ]\left [-\frac {x^2}{2}+\log (x)+c_1\right ]\\ y(x)&\to -i\\ y(x)&\to i \end{align*}

✓ Sympy. Time used: 1.642 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x)**3*Derivative(y(x), x) - (1 - x**2)*(y(x)**2 + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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