23.3.46 problem 48

Internal problem ID [5760]
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 : 48
Date solved : Friday, October 03, 2025 at 01:43:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\operatorname {a0} +\operatorname {a1} \cos \left (x \right )^{2}+\operatorname {a2} \csc \left (x \right )^{2}\right ) y+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.167 (sec). Leaf size: 80

ode:=(a0+a1*cos(x)^2+a2*csc(x)^2)*y(x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (x \right )^{\frac {1}{2}+\frac {\sqrt {-4 \operatorname {a2} +1}}{2}} \left (c_1 \operatorname {HeunC}\left (0, -\frac {1}{2}, \frac {\sqrt {-4 \operatorname {a2} +1}}{2}, -\frac {\operatorname {a1}}{4}, \frac {3}{8}-\frac {\operatorname {a0}}{4}-\frac {\operatorname {a2}}{4}, \cos \left (x \right )^{2}\right )+c_2 \operatorname {HeunC}\left (0, \frac {1}{2}, \frac {\sqrt {-4 \operatorname {a2} +1}}{2}, -\frac {\operatorname {a1}}{4}, \frac {3}{8}-\frac {\operatorname {a0}}{4}-\frac {\operatorname {a2}}{4}, \cos \left (x \right )^{2}\right ) \cos \left (x \right )\right ) \]

✗ Mathematica

ode=(a0 + a1*Cos[x]^2 + a2*Csc[x]^2)*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") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
y = Function("y") 
ode = Eq((a0 + a1*cos(x)**2 + a2/sin(x)**2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve (a0 + a1*cos(x)**2 + a2/sin(x)**2)*y(x) + Derivative(y(x),