54.3.63 problem 1068

Internal problem ID [12358]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1068
Date solved : Friday, October 03, 2025 at 03:18:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime } \cot \left (x \right )+v \left (v +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.112 (sec). Leaf size: 45

ode:=diff(diff(y(x),x),x)+diff(y(x),x)*cot(x)+v*(v+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.092 (sec). Leaf size: 20

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

\begin{align*} y(x)&\to c_1 \operatorname {LegendreP}(v,\cos (x))+c_2 \operatorname {LegendreQ}(v,\cos (x)) \end{align*}

✗ Sympy

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

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