20.22.14 problem Problem 40

Internal problem ID [3969]
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 40
Date solved : Monday, January 27, 2025 at 08:05:24 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&=5 \operatorname {Heaviside}\left (t -3\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 4.976 (sec). Leaf size: 47

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+5*y(t)=5*Heaviside(t-3),y(0) = 2, D(y)(0) = 1],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 68

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+5*y[t]==5*UnitStep[t-3],{y[0]==2,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-2 t} (2 \cos (t)+5 \sin (t)) & t\leq 3 \\ e^{-2 t} \left (-e^6 \cos (3-t)+e^{2 t}+2 \cos (t)+2 e^6 \sin (3-t)+5 \sin (t)\right ) & \text {True} \\ \end {array} \\ \end {array} \]