28.3.6 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 : Monday, January 27, 2025 at 09:22:38 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*}

✓ Solution by Maple

Time used: 3.331 (sec). Leaf size: 22

dsolve([diff(y(t),t$2)-2*diff(y(t),t)+2*y(t)=8*exp(-t)*sin(t),y(0) = 1, D(y)(0) = -1],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 25

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

\[ y(t)\to e^{-t} \left (-e^{2 t} \sin (t)+\sin (t)+\cos (t)\right ) \]