54.3.399 problem 1416

Internal problem ID [12694]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1416
Date solved : Friday, October 03, 2025 at 03:46:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.074 (sec). Leaf size: 26

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

\[ y = \sin \left (x \right )^{-n} \left (c_1 \operatorname {LegendreP}\left (v , n , \cos \left (x \right )\right )+c_2 \operatorname {LegendreQ}\left (v , n , \cos \left (x \right )\right )\right ) \]

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 35

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

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

NotImplementedError : The given ODE Derivative(y(x), x) - (n**2*y(x) + n*y(x) - v**2*y(x) - v*y(x) - Derivative(y(x), (x, 2)))*tan(x)/(2*n + 1) cannot be solved by the factorable group method