28.3.11 problem 6.46

Internal problem ID [4524]
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 : 6.46
Date solved : Tuesday, March 04, 2025 at 06:51:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=8 \sin \left (2 t \right ) \operatorname {Heaviside}\left (t -\pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 3.299 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)+4*y(t) = 8*sin(2*t)*Heaviside(t-Pi); 
ic:=y(0) = 0, D(y)(0) = 2; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \left (\left (-2 t +2 \pi \right ) \cos \left (2 t \right )+\sin \left (2 t \right )\right ) \operatorname {Heaviside}\left (t -\pi \right )+\sin \left (2 t \right ) \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 32

ode=D[y[t],{t,2}]+4*y[t]==8*Sin[2*t]*UnitStep[t-Pi]; 
ic={y[0]==0,Derivative[1][y][0] == 2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \sin (2 t) & t\leq \pi \\ 2 ((\pi -t) \cos (2 t)+\sin (2 t)) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 1.180 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 8*sin(2*t)*Heaviside(t - pi) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

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