29.24.28 problem 691

Internal problem ID [5281]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 691
Date solved : Sunday, March 30, 2025 at 07:50:37 AM
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: 60.102 (sec). Leaf size: 61

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

✓ Sympy. Time used: 2.739 (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)
 

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