Internal problem ID [8649]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1071.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+2 a y^{\prime } \cot \left (a x \right )+\left (-a^{2}+b^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 37
dsolve(diff(diff(y(x),x),x)+2*a*diff(y(x),x)*cot(a*x)+(-a^2+b^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {\cot \left (a x \right )^{2}+1}\, \sin \left (x b \right )+c_{2} \sqrt {\cot \left (a x \right )^{2}+1}\, \cos \left (x b \right ) \]
✓ Solution by Mathematica
Time used: 0.06 (sec). Leaf size: 43
DSolve[(-a^2 + b^2)*y[x] + 2*a*Cot[a*x]*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{-i b x} \csc (a x) \left (2 c_1-\frac {i c_2 e^{2 i b x}}{b}\right ) \\ \end{align*}