60.3.70 problem 1071

Internal problem ID [11080]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1071
Date solved : Monday, January 27, 2025 at 10:44:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 22

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)
 

\[ y = \csc \left (a x \right ) \left (c_{1} \sin \left (b x \right )+c_{2} \cos \left (b x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.093 (sec). Leaf size: 43

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

\[ y(x)\to \frac {1}{2} e^{-i b x} \csc (a x) \left (2 c_1-\frac {i c_2 e^{2 i b x}}{b}\right ) \]