54.3.419 problem 1436

Internal problem ID [12714]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1436
Date solved : Friday, October 03, 2025 at 03:47:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (4 v \left (v +1\right ) \sin \left (x \right )^{2}-\cos \left (x \right )^{2}+2-4 n^{2}\right ) y}{4 \sin \left (x \right )^{2}} \end{align*}

✓ Maple. Time used: 0.228 (sec). Leaf size: 91

ode:=diff(diff(y(x),x),x) = -1/4*(4*v*(v+1)*sin(x)^2-cos(x)^2+2-4*n^2)/sin(x)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.394 (sec). Leaf size: 33

ode=D[y[x],{x,2}] == -1/4*(Csc[x]^2*(2 - 4*n^2 - Cos[x]^2 + 4*v*(1 + v)*Sin[x]^2)*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt [4]{-\sin ^2(x)} (c_1 P_v^n(\cos (x))+c_2 Q_v^n(\cos (x))) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
n = symbols("n") 
v = symbols("v") 
y = Function("y") 
ode = Eq((-4*n**2 + 4*v*(v + 1)*sin(x)**2 - cos(x)**2 + 2)*y(x)/(4*sin(x)**2) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve (-4*n**2 + 4*v*(v + 1)*sin(x)**2 - cos(x)**2 + 2)*y(x)/(4*sin(x)**2) + Derivative(y(x), (x, 2))