82.54.12 problem Ex. 12

Internal problem ID [18949]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at end of chapter at page 120
Problem number : Ex. 12
Date solved : Thursday, March 13, 2025 at 01:13:06 PM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }-a^{2} y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{a \arcsin \left (x \right )} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 42

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

\[ y(x)\to c_1 \cos \left (a \log \left (\sqrt {x^2-1}+x\right )\right )+c_2 \sin \left (a \log \left (\sqrt {x^2-1}+x\right )\right ) \]

✗ Sympy

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

False