11.9 problem 2(c)

Internal problem ID [5572]

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).
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]

Solve \begin {gather*} \boxed {y^{\prime \prime }+y-\cot \left (2 x \right )=0} \end {gather*}

Solution by Maple

Time used: 0.003 (sec). Leaf size: 43

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

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

Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 45

DSolve[y''[x]+y[x]==Cot[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{2} \left (\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 )+\cos (x) \left (\tanh ^{-1}(\sin (x))+2 c_1\right )\right ) \\ \end{align*}