28.3.9 problem 6.44

Internal problem ID [4522]
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 : 6.44
Date solved : Monday, January 27, 2025 at 09:22:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y^{\prime }-2 y&=9 \,{\mathrm e}^{2 t} \operatorname {Heaviside}\left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=6\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 3.521 (sec). Leaf size: 42

dsolve([diff(y(t),t$2)-diff(y(t),t)-2*y(t)=9*exp(2*t)*Heaviside(t-1),y(0) = 6, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \operatorname {Heaviside}\left (-1+t \right ) {\mathrm e}^{-t +3}+{\mathrm e}^{2 t} \left (-4+3 t \right ) \operatorname {Heaviside}\left (-1+t \right )+4 \,{\mathrm e}^{-t}+2 \,{\mathrm e}^{2 t} \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 47

DSolve[{D[y[t],{t,2}]-D[y[t],t]-2*y[t]==9*Exp[2*t]*UnitStep[t-1],{y[0]==6,Derivative[1][y][0] == 0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 2 e^{-t} \left (2+e^{3 t}\right ) & t\leq 1 \\ e^{-t} \left (e^{3 t} (3 t-2)+e^3+4\right ) & \text {True} \\ \end {array} \\ \end {array} \]