12.2.8 problem 8

Internal problem ID [1544]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Linear first order. Section 2.1 Page 41
Problem number : 8
Date solved : Monday, January 27, 2025 at 04:59:29 AM
CAS classification : [_separable]

\begin{align*} x y^{\prime }+\left (1+x \cot \left (x \right )\right ) y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&=2 \end{align*}

Solution by Maple

Time used: 0.007 (sec). Leaf size: 11

dsolve([x*diff(y(x),x) + (1+x*cot(x))*y(x)=0,y(1/2*Pi) = 2],y(x), singsol=all)
 
\[ y = \frac {\csc \left (x \right ) \pi }{x} \]

Solution by Mathematica

Time used: 0.091 (sec). Leaf size: 66

DSolve[{D[y[x],x] +(1+x*Cot[x])*y[x]==0,y[Pi/2]==2},y[x],x,IncludeSingularSolutions -> True]
 
\[ y(x)\to 2^{1+\frac {\pi }{2}} \left (1-e^{2 i x}\right )^{-x} \exp \left (-\frac {1}{12} i \left (-6 \operatorname {PolyLog}\left (2,e^{2 i x}\right )-6 x (x+2 i)+\pi ^2+6 i \pi \right )\right ) \]