78.14.11 problem 4 (c)

Internal problem ID [18276]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 19. The Method of Variation of Parameters. Problems at page 135
Problem number : 4 (c)
Date solved : Monday, March 31, 2025 at 05:24:34 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (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: 0.061 (sec). Leaf size: 46

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

\[ y(x)\to \frac {1}{2} \left (\cos (x) \text {arctanh}(\sin (x))+2 c_1 \cos (x)+\sin (x) \left (\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right )+2 c_2\right )\right ) \]

✓ Sympy. Time used: 0.773 (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)
 

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