54.3.25 problem 1025

Internal problem ID [12320]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1025
Date solved : Friday, October 03, 2025 at 03:18:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\left (\frac {m \left (m -1\right )}{\cos \left (x \right )^{2}}+\frac {n \left (n -1\right )}{\sin \left (x \right )^{2}}+a \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.159 (sec). Leaf size: 98

ode:=diff(diff(y(x),x),x)-(m*(m-1)/cos(x)^2+n*(n-1)/sin(x)^2+a)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.962 (sec). Leaf size: 158

ode=(-a - (-1 + n)*n*Csc[x]^2 - (-1 + m)*m*Sec[x]^2)*y[x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

NotImplementedError : solve: Cannot solve (-a - m*(m - 1)/cos(x)**2 - n*(n - 1)/sin(x)**2)*y(x) + Derivative(y(x), (x, 2))