83.37.11 problem 11

Internal problem ID [19456]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (B) at page 128
Problem number : 11
Date solved : Tuesday, January 28, 2025 at 01:44:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 22

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

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

✓ Solution by Mathematica

Time used: 0.081 (sec). Leaf size: 43

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

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