4.1.25 problem 25

Internal problem ID [1122]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Section 2.1. Page 40
Problem number : 25
Date solved : Tuesday, September 30, 2025 at 04:22:25 AM
CAS classification : [_linear]

\begin{align*} 2 y+t y^{\prime }&=\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: 0.033 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 22

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

\begin{align*} y(t)&\to \frac {\pi ^2 a-4 \cos (t)}{4 t^2} \end{align*}

✓ Sympy. Time used: 0.289 (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)
 

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