60.3.8 problem 1008

Internal problem ID [11018]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1008
Date solved : Monday, January 27, 2025 at 10:41:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+a^{2} y-\cot \left (a x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 37

dsolve(diff(diff(y(x),x),x)+a^2*y(x)-cot(a*x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.082 (sec). Leaf size: 68

DSolve[-Cot[a*x] + a^2*y[x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \cos (a x) \int _1^x-\frac {\cos (a K[1])}{a}dK[1]+\sin (a x) \int _1^x\frac {\cos (a K[2]) \cot (a K[2])}{a}dK[2]+c_1 \cos (a x)+c_2 \sin (a x) \]