54.3.407 problem 1424

Internal problem ID [12702]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1424
Date solved : Friday, October 03, 2025 at 03:46:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.221 (sec). Leaf size: 98

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

\[ y = \frac {\sqrt {\cos \left (x \right )}\, \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {1}{4}+\frac {n}{2}} \left (\cos \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {n}{2}+\frac {1}{2}+\frac {i \sqrt {a}}{2}, \frac {n}{2}+\frac {1}{2}-\frac {i \sqrt {a}}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_2 +\operatorname {hypergeom}\left (\left [\frac {n}{2}+\frac {i \sqrt {a}}{2}, \frac {n}{2}-\frac {i \sqrt {a}}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_1 \right )}{\sqrt {\sin \left (2 x \right )}} \]

✓ Mathematica. Time used: 0.169 (sec). Leaf size: 65

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

\begin{align*} y(x)&\to \sqrt [4]{-\sin ^2(x)} \left (c_1 P_{i \sqrt {a}-\frac {1}{2}}^{n-\frac {1}{2}}(\cos (x))+c_2 Q_{i \sqrt {a}-\frac {1}{2}}^{n-\frac {1}{2}}(\cos (x))\right ) \end{align*}

✗ Sympy

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

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