28.3.10 problem 6.45

Internal problem ID [4523]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 6. The Laplace Transform and Its Applications. Problems at page 291
Problem number : 6.45
Date solved : Tuesday, March 04, 2025 at 06:51:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 3.316 (sec). Leaf size: 38

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+y(t) = 2*sin(t)*Heaviside(t-Pi); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \left (\left (-t +\pi -1\right ) {\mathrm e}^{-t +\pi }-\cos \left (t \right )\right ) \operatorname {Heaviside}\left (t -\pi \right )+{\mathrm e}^{-t} \left (t +1\right ) \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 46

ode=D[y[t],{t,2}]+2*D[y[t],t]+y[t]==2*Sin[t]*UnitStep[t-Pi]; 
ic={y[0]==1,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} (t+1) & t\leq \pi \\ e^{-t} \left (e^{\pi } (-t+\pi -1)+t-e^t \cos (t)+1\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 1.553 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - 2*sin(t)*Heaviside(t - pi) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (t \left (- e^{\pi } \theta \left (t - \pi \right ) + 1\right ) - e^{\pi } \theta \left (t - \pi \right ) + \pi e^{\pi } \theta \left (t - \pi \right ) + 1\right ) e^{- t} - \cos {\left (t \right )} \theta \left (t - \pi \right ) \]