23.3.381 problem 385

Internal problem ID [6095]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 385
Date solved : Friday, October 03, 2025 at 01:46:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.125 (sec). Leaf size: 63

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

\[ y = c_1 \operatorname {MathieuC}\left (\frac {a^{2} c^{2}}{2}+b^{2}, -\frac {a^{2} c^{2}}{4}, \arccos \left (\frac {x}{a}\right )\right )+c_2 \operatorname {MathieuS}\left (\frac {a^{2} c^{2}}{2}+b^{2}, -\frac {a^{2} c^{2}}{4}, \arccos \left (\frac {x}{a}\right )\right ) \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 74

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

\begin{align*} y(x)&\to c_1 \text {MathieuC}\left [\frac {a^2 c^2}{2}+b^2,-\frac {1}{4} a^2 c^2,\arccos \left (\frac {x}{a}\right )\right ]+c_2 \text {MathieuS}\left [\frac {a^2 c^2}{2}+b^2,-\frac {1}{4} a^2 c^2,\arccos \left (\frac {x}{a}\right )\right ] \end{align*}

✗ Sympy

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

False