20.22.13 problem Problem 39

Internal problem ID [3968]
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 10.7. page 704
Problem number : Problem 39
Date solved : Monday, January 27, 2025 at 08:05:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }-6 y&=30 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-t +1} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=-4 \end{align*}

✓ Solution by Maple

Time used: 3.279 (sec). Leaf size: 47

dsolve([diff(y(t),t$2)+diff(y(t),t)-6*y(t)=30*Heaviside(t-1)*exp(-(t-1)),y(0) = 3, D(y)(0) = -4],y(t), singsol=all)
 

\[ y = \left (3 \operatorname {Heaviside}\left (-1+t \right ) {\mathrm e}^{3}+{\mathrm e}^{5 t}+2 \,{\mathrm e}^{-2+5 t} \operatorname {Heaviside}\left (-1+t \right )-5 \operatorname {Heaviside}\left (-1+t \right ) {\mathrm e}^{1+2 t}+2\right ) {\mathrm e}^{-3 t} \]

✓ Solution by Mathematica

Time used: 0.088 (sec). Leaf size: 66

DSolve[{D[y[t],{t,2}]+D[y[t],t]-6*y[t]==30*UnitStep[t-1]*Exp[-(t-1)],{y[0]==3,Derivative[1][y][0] ==-4}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-3 t} \left (2+e^{5 t}\right ) & t\leq 1 \\ e^{-3 t-2} \left (2 e^2+3 e^5+2 e^{5 t}-5 e^{2 t+3}+e^{5 t+2}\right ) & \text {True} \\ \end {array} \\ \end {array} \]