76.2.25 problem 25

Internal problem ID [17290]
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 : Monday, March 31, 2025 at 03:48:59 PM
CAS classification : [_linear]

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

With initial conditions

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 19

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

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

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

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

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

✓ Sympy. Time used: 0.372 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), t) + 2*y(t) - sin(t)/t,0) 
ics = {y(-pi/2): a} 
dsolve(ode,func=y(t),ics=ics)
 

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