28.3.13 problem 6.48

Internal problem ID [4526]
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 : 6.48
Date solved : Monday, January 27, 2025 at 09:22:45 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+2 y&={\mathrm e}^{t} \operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 3.495 (sec). Leaf size: 31

dsolve([diff(y(t),t$2)-3*diff(y(t),t)+2*y(t)=exp(t)*Heaviside(t-2),y(0) = 1, D(y)(0) = 2],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 33

DSolve[{D[y[t],{t,2}]-3*D[y[t],t]+2*y[t]==Exp[t]*UnitStep[t-2],{y[0]==1,Derivative[1][y][0] == 2}},y[t],t,IncludeSingularSolutions -> True]
 

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