22.3.6 problem 6.41

Internal problem ID [4519]
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 : 6.41
Date solved : Tuesday, September 30, 2025 at 07:33:48 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+2 y&=8 \,{\mathrm e}^{-t} \sin \left (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: 0.276 (sec). Leaf size: 22

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

\[ y = \cosh \left (t \right ) \cos \left (t \right )+\left (-\cos \left (t \right )-2 \sin \left (t \right )\right ) \sinh \left (t \right ) \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 25

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

\begin{align*} y(t)&\to e^{-t} \left (-e^{2 t} \sin (t)+\sin (t)+\cos (t)\right ) \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 19

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

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