20.23.5 problem Problem 5

Internal problem ID [3977]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.8. page 710
Problem number : Problem 5
Date solved : Monday, January 27, 2025 at 08:05:33 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+2 y&=\delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 3.408 (sec). Leaf size: 36

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

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

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 31

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

\[ y(t)\to e^t \left (\frac {\left (e^t-e\right ) \theta (t-1)}{e^2}-e^t+2\right ) \]