54.3.8 problem 1008

Internal problem ID [12303]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1008
Date solved : Wednesday, October 01, 2025 at 01:42:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 37

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

\[ y = \sin \left (a x \right ) c_2 +\cos \left (a x \right ) c_1 +\frac {\sin \left (a x \right ) \ln \left (\csc \left (a x \right )-\cot \left (a x \right )\right )}{a^{2}} \]

✓ Mathematica. Time used: 0.052 (sec). Leaf size: 68

ode=-Cot[a*x] + a^2*y[x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \cos (a x) \int _1^x-\frac {\cos (a K[1])}{a}dK[1]+\sin (a x) \int _1^x\frac {\cos (a K[2]) \cot (a K[2])}{a}dK[2]+c_1 \cos (a x)+c_2 \sin (a x) \end{align*}

✓ Sympy. Time used: 0.394 (sec). Leaf size: 124

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

\[ y{\left (x \right )} = \left (C_{1} - \frac {i \left (\begin {cases} \frac {e^{- i a x}}{a} & \text {for}\: a \neq 0 \\- i x & \text {otherwise} \end {cases}\right )}{2 a} - \frac {i \log {\left (e^{i a x} - 1 \right )}}{2 a^{2}} + \frac {i \log {\left (e^{i a x} + 1 \right )}}{2 a^{2}}\right ) e^{i a x} + \left (C_{2} + \frac {i \left (\begin {cases} \frac {e^{i a x}}{a} & \text {for}\: a \neq 0 \\i x & \text {otherwise} \end {cases}\right )}{2 a} + \frac {i \log {\left (e^{i a x} - 1 \right )}}{2 a^{2}} - \frac {i \log {\left (e^{i a x} + 1 \right )}}{2 a^{2}}\right ) e^{- i a x} \]