28.3.7 problem 6.42

Internal problem ID [4520]
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 : 6.42
Date solved : Tuesday, March 04, 2025 at 06:50:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&=8 \,{\mathrm e}^{t} \sin \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 3.725 (sec). Leaf size: 16

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+5*y(t) = 8*exp(t)*sin(2*t); 
ic:=y(0) = 1, D(y)(0) = -1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \left (-2 t +1\right ) {\mathrm e}^{t} \cos \left (2 t \right ) \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 19

ode=D[y[t],{t,2}]-2*D[y[t],t]+5*y[t]==8*Exp[t]*Sin[2*t]; 
ic={y[0]==1,Derivative[1][y][0] == -1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -e^t (2 t-1) \cos (2 t) \]

✓ Sympy. Time used: 0.289 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) - 8*exp(t)*sin(2*t) - 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (1 - 2 t\right ) e^{t} \cos {\left (2 t \right )} \]