23.3.356 problem 359

Internal problem ID [6070]
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 : 359
Date solved : Friday, October 03, 2025 at 01:46:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 42

ode=(a + b*x^2)*y[x] - x*D[y[x],x] + (1 - 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 [a+\frac {b}{2},-\frac {b}{4},\arccos (x)\right ]+c_2 \text {MathieuS}\left [a+\frac {b}{2},-\frac {b}{4},\arccos (x)\right ] \end{align*}

✗ Sympy

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

False