54.3.66 problem 1071

Internal problem ID [12361]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1071
Date solved : Wednesday, October 01, 2025 at 01:43:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 22

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

\[ y = \csc \left (a x \right ) \left (c_1 \sin \left (b x \right )+c_2 \cos \left (b x \right )\right ) \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 43

ode=(-a^2 + b^2)*y[x] + 2*a*Cot[a*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 \frac {1}{2} e^{-i b x} \csc (a x) \left (2 c_1-\frac {i c_2 e^{2 i b x}}{b}\right ) \end{align*}

✗ Sympy

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

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