23.3.43 problem 44

Internal problem ID [5757]
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 : 44
Date solved : Friday, October 03, 2025 at 01:43:41 AM
CAS classification : [_ellipsoidal]

\begin{align*} \left (a +b \cos \left (2 x \right )+k \cos \left (4 x \right )\right ) y+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.192 (sec). Leaf size: 82

ode:=(a+b*cos(2*x)+k*cos(4*x))*y(x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\sqrt {2}\, \sqrt {k}\, \cos \left (x \right )^{2}} \left (c_1 \operatorname {HeunC}\left (2 \sqrt {2}\, \sqrt {k}, -\frac {1}{2}, -\frac {1}{2}, -\frac {b}{2}, \frac {3}{8}-\frac {a}{4}+\frac {b}{4}-\frac {k}{4}, \cos \left (x \right )^{2}\right )+c_2 \operatorname {HeunC}\left (2 \sqrt {2}\, \sqrt {k}, \frac {1}{2}, -\frac {1}{2}, -\frac {b}{2}, \frac {3}{8}-\frac {a}{4}+\frac {b}{4}-\frac {k}{4}, \cos \left (x \right )^{2}\right ) \cos \left (x \right )\right ) \]

✗ Mathematica

ode=(a + b*Cos[2*x] + k*Cos[4*x])*y[x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
k = symbols("k") 
y = Function("y") 
ode = Eq((a + b*cos(2*x) + k*cos(4*x))*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False