76.2.18 problem 18

Internal problem ID [17283]
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 : 18
Date solved : Monday, March 31, 2025 at 03:48:42 PM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&=3 \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 22

ode:=t*diff(y(t),t)+2*y(t) = sin(t); 
ic:=y(1/2*Pi) = 3; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {\sin \left (t \right )-\cos \left (t \right ) t +\frac {3 \pi ^{2}}{4}-1}{t^{2}} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 37

ode=t*D[y[t],t]+2*y[t]==Sin[t]; 
ic={y[Pi/2]==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {4 \int _{\frac {\pi }{2}}^tK[1] \sin (K[1])dK[1]+3 \pi ^2}{4 t^2} \]

✓ Sympy. Time used: 0.307 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), t) + 2*y(t) - sin(t),0) 
ics = {y(pi/2): 3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {- \cos {\left (t \right )} + \frac {\sin {\left (t \right )}}{t} + \frac {-1 + \frac {3 \pi ^{2}}{4}}{t}}{t} \]