23.3.47 problem 49

Internal problem ID [5761]
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 : 49
Date solved : Friday, October 03, 2025 at 01:43:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\left (a^{2}+\left (-1+p \right ) p \csc \left (x \right )^{2}+\left (-1+q \right ) q \sec \left (x \right )^{2}\right ) y \end{align*}

✓ Maple. Time used: 0.145 (sec). Leaf size: 94

ode:=diff(diff(y(x),x),x) = (a^2+(-1+p)*p*csc(x)^2+(-1+q)*q*sec(x)^2)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (x \right )^{p} \left (c_1 \cos \left (x \right )^{q} \operatorname {hypergeom}\left (\left [\frac {p}{2}+\frac {q}{2}+\frac {i a}{2}, \frac {p}{2}+\frac {q}{2}-\frac {i a}{2}\right ], \left [\frac {1}{2}+q \right ], \cos \left (x \right )^{2}\right )+c_2 \cos \left (x \right )^{1-q} \operatorname {hypergeom}\left (\left [\frac {p}{2}-\frac {q}{2}+\frac {i a}{2}+\frac {1}{2}, \frac {p}{2}-\frac {q}{2}-\frac {i a}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}-q \right ], \cos \left (x \right )^{2}\right )\right ) \]

✓ Mathematica. Time used: 0.764 (sec). Leaf size: 146

ode=D[y[x],{x,2}] == (a^2 + (-1 + p)*p*Csc[x]^2 + (-1 + q)*q*Sec[x]^2)*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {(-1)^{-q} \left (-\sin ^2(x)\right )^{p/2} \cos ^2(x)^{-\frac {q}{2}-\frac {1}{4}} \left (c_1 (-1)^q \cos ^2(x)^{q+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-i a+p+q),\frac {1}{2} (i a+p+q),q+\frac {1}{2},\cos ^2(x)\right )+i c_2 \cos ^2(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-i a+p-q+1),\frac {1}{2} (i a+p-q+1),\frac {3}{2}-q,\cos ^2(x)\right )\right )}{\sqrt {\cos (x)}} \end{align*}

✗ Sympy

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

NotImplementedError : solve: Cannot solve (-a**2 - p*(p - 1)/sin(x)**2 - q*(q - 1)/cos(x)**2)*y(x) +