87.8.19 problem 19

Internal problem ID [23373]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 65
Problem number : 19
Date solved : Friday, October 03, 2025 at 08:03:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.210 (sec). Leaf size: 84

ode:=(1+a*cos(2*x))*diff(diff(y(x),x),x)+lambda*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {HeunG}\left (\frac {2 a}{a -1}, \frac {\lambda }{4 a -4}, 0, 0, \frac {1}{2}, 0, \frac {a \left (1+\cos \left (2 x \right )\right )}{a -1}\right )+c_2 \operatorname {HeunG}\left (\frac {2 a}{a -1}, \frac {\lambda +a -1}{4 a -4}, \frac {1}{2}, \frac {1}{2}, \frac {3}{2}, 0, \frac {a \left (1+\cos \left (2 x \right )\right )}{a -1}\right ) \cos \left (x \right ) \]

✗ Mathematica

ode=(1+a*Cos[2*x])*D[y[x],{x,2}]+\[Lambda]*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

False