19.1.13 problem 13

Internal problem ID [3527]
Book : Differential equations and linear algebra, Stephen W. Goode, second edition, 2000
Section : 1.4, page 36
Problem number : 13
Date solved : Sunday, March 30, 2025 at 01:46:05 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=2 a \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 20

ode:=(-x^2+1)*diff(y(x),x)+x*y(x) = a*x; 
ic:=y(0) = 2*a; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = a \left (1-i \sqrt {x -1}\, \sqrt {x +1}\right ) \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 21

ode=(1-x^2)*D[y[x],x]+x*y[x]==a*x; 
ic=y[0]==2*a; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to a-i a \sqrt {x^2-1} \]

✓ Sympy. Time used: 0.272 (sec). Leaf size: 15

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

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