76.2.25 problem 25

Internal problem ID [17361]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.2 (Linear equations: Method of integrating factors). Problems at page 54
Problem number : 25
Date solved : Tuesday, January 28, 2025 at 10:02:25 AM
CAS classification : [_linear]

\begin{align*} y^{\prime } t +2 y&=\frac {\sin \left (t \right )}{t} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 19

dsolve([t*diff(y(t),t)+2*y(t)=sin(t)/t,y(-1/2*Pi) = a],y(t), singsol=all)
 

\[ y = \frac {-\cos \left (t \right )+\frac {a \,\pi ^{2}}{4}}{t^{2}} \]

✓ Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 34

DSolve[{t*D[y[t],t]+2*y[t]==Sin[t]/t,{y[-Pi/2]==a}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {4 \int _{-\frac {\pi }{2}}^t\sin (K[1])dK[1]+\pi ^2 a}{4 t^2} \]