44.11.9 problem 2(c)

Internal problem ID [9279]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.3. THE METHOD OF VARIATION OF PARAMETERS. Page 71
Problem number : 2(c)
Date solved : Tuesday, September 30, 2025 at 06:16:02 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 50

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

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

✓ Sympy. Time used: 0.503 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = \left (C_{1} - \frac {\log {\left (\sin {\left (x \right )} - 1 \right )}}{4} + \frac {\log {\left (\sin {\left (x \right )} + 1 \right )}}{4}\right ) \cos {\left (x \right )} + \left (C_{2} + \frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{4} - \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{4}\right ) \sin {\left (x \right )} \]