23.3.42 problem 43

Internal problem ID [5756]
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 : 43
Date solved : Friday, October 03, 2025 at 01:43:40 AM
CAS classification : [_ellipsoidal]

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

✓ Maple. Time used: 0.079 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 28

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

✗ Sympy

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

False