10.1.18 problem 18

Internal problem ID [1115]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Section 2.1. Page 40
Problem number : 18
Date solved : Saturday, March 29, 2025 at 10:39:31 PM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&=1 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 26

ode=2*y[t]+t*D[y[t],t] == Sin[t]; 
ic=y[Pi/2]==1; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {4 \sin (t)-4 t \cos (t)+\pi ^2-4}{4 t^2} \]

✓ Sympy. Time used: 0.299 (sec). Leaf size: 20

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): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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