78.14.11 problem 4 (c)

Internal problem ID [18347]
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 : Tuesday, January 28, 2025 at 11:47:53 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 35

dsolve(diff(y(x),x$2)+y(x)=cot(2*x),y(x), singsol=all)
 

\[ y = \sin \left (x \right ) c_{2} +\cos \left (x \right ) c_{1} +\frac {\sin \left (x \right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{2}+\frac {\cos \left (x \right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.061 (sec). Leaf size: 46

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

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