23.3.31 problem 31

Internal problem ID [5745]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 31
Date solved : Tuesday, September 30, 2025 at 02:02:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 37

ode:=a^2*y(x)+diff(diff(y(x),x),x) = cot(a*x); 
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: 0.047 (sec). Leaf size: 46

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

\begin{align*} y(x)&\to \frac {\sin (a x) \left (a^2 c_2+\log \left (\sin \left (\frac {a x}{2}\right )\right )-\log \left (\cos \left (\frac {a x}{2}\right )\right )\right )}{a^2}+c_1 \cos (a x) \end{align*}

✓ Sympy. Time used: 0.387 (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)
 

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