23.1.393 problem 378

Internal problem ID [5000]
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 : 378
Date solved : Tuesday, September 30, 2025 at 11:15:44 AM
CAS classification : [_rational, _Bernoulli]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 72

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

\[ y = \frac {\sqrt {x^{2}-1}\, x}{\sqrt {x +1}\, \sqrt {x -1}\, c_1 \sqrt {x^{2}-1}-3 \ln \left (x +\sqrt {x^{2}-1}\right ) x^{2}+3 \sqrt {x^{2}-1}\, x +3 \ln \left (x +\sqrt {x^{2}-1}\right )} \]

✓ Mathematica. Time used: 0.218 (sec). Leaf size: 78

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

\begin{align*} y(x)&\to \frac {\exp \left (\int _1^x\frac {1}{K[1]-K[1]^3}dK[1]\right )}{-\int _1^x\frac {3 \exp \left (\int _1^{K[2]}\frac {1}{K[1]-K[1]^3}dK[1]\right ) K[2]}{K[2]^2-1}dK[2]+c_1}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.307 (sec). Leaf size: 51

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

\[ y{\left (x \right )} = \frac {x}{C_{1} \sqrt {x^{2} - 1} + 3 x - 3 \sqrt {x^{2} - 1} \log {\left (x + \sqrt {x^{2} - 1} \right )} - \sqrt {x^{2} - 1} \log {\left (8 \right )}} \]