29.13.21 problem 375

Internal problem ID [4975]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 13
Problem number : 375
Date solved : Sunday, March 30, 2025 at 04:27:34 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: 0.276 (sec). Leaf size: 59

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 {x}{-3 \sqrt {1-x^2} \arctan \left (\frac {x}{\sqrt {1-x^2}}\right )+c_1 \sqrt {1-x^2}+3 x} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.438 (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)
 

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