29.11.9 problem 300

Internal problem ID [4900]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 11
Problem number : 300
Date solved : Sunday, March 30, 2025 at 04:09:43 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 22

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

\[ y = \frac {1}{\sqrt {x -1}\, \sqrt {x +1}\, c_1 -a} \]

✓ Mathematica. Time used: 3.833 (sec). Leaf size: 47

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

\begin{align*} y(x)\to -\frac {e^{c_1}}{-\sqrt {1-x^2}+a e^{c_1}} \\ y(x)\to 0 \\ y(x)\to -\frac {1}{a} \\ \end{align*}

✓ Sympy. Time used: 1.599 (sec). Leaf size: 44

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

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