82.54.15 problem Ex. 15

Internal problem ID [18960]
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. 15
Date solved : Monday, March 31, 2025 at 06:26:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 69

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

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

✗ Sympy

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

False