28.3.14 problem 6.49

Internal problem ID [4527]
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 : 6.49
Date solved : Monday, January 27, 2025 at 09:22:46 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }+6 y&=\delta \left (t -2\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: 3.950 (sec). Leaf size: 40

dsolve([diff(y(t),t$2)-5*diff(y(t),t)+6*y(t)=Dirac(t-2),y(0) = -1, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = -\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2 t -4}+\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-6+3 t}-4 \,{\mathrm e}^{2 t}+3 \,{\mathrm e}^{3 t} \]

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 39

DSolve[{D[y[t],{t,2}]-5*D[y[t],t]+6*y[t]==DiracDelta[t-2],{y[0]==-1,Derivative[1][y][0] == 1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{2 t-6} \left (\left (e^t-e^2\right ) \theta (t-2)+e^6 \left (3 e^t-4\right )\right ) \]