Internal problem ID [8588]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1008.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+a^{2} y-\cot \left (a x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 41
dsolve(diff(diff(y(x),x),x)+a^2*y(x)-cot(a*x)=0,y(x), singsol=all)
\[ y \relax (x ) = \sin \left (a x \right ) c_{2}+\cos \left (a x \right ) c_{1}+\frac {\sin \left (a x \right ) \ln \left (\frac {1-\cos \left (a x \right )}{\sin \left (a x \right )}\right )}{a^{2}} \]
✓ Solution by Mathematica
Time used: 0.03 (sec). Leaf size: 46
DSolve[-Cot[a*x] + a^2*y[x] + y''[x] == 0,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*}