54.3.413 problem 1430

Internal problem ID [12708]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1430
Date solved : Friday, October 03, 2025 at 03:46:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.229 (sec). Leaf size: 79

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

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

✓ Mathematica. Time used: 0.302 (sec). Leaf size: 22

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

\begin{align*} y(x)&\to 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((-n**2 + v*(v + 1)*sin(x)**2)*y(x)/sin(x)**2 + Derivative(y(x), (x, 2)) + cos(x)*Derivative(y(x), x)/sin(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(n**2*y(x) - (v**2*y(x) + v*y(x) + Derivative(y(x), (x, 2)))*sin(x)**2)/(sin(x)*cos(x)) + Derivative(y(x), x) cannot be solved by the factorable group method