Internal problem ID [2371]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+4 y^{\prime }+5 y-5 \theta \left (t -3\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 2, y^{\prime }\relax (0) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.044 (sec). Leaf size: 46
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)
\[ y \relax (t ) = -\theta \left (t -3\right ) \left (2 \sin \left (t -3\right )+\cos \left (t -3\right )\right ) {\mathrm e}^{-2 t +6}+\theta \left (t -3\right )+\left (2 \cos \relax (t )+5 \sin \relax (t )\right ) {\mathrm e}^{-2 t} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 63
DSolve[{y''[t]+4*y'[t]+5*y[t]==5*UnitStep[t-3],{y[0]==2,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} e^{-2 t} (2 \cos (t)+5 \sin (t)) & t\leq 3 \\ e^{-2 t} \left (2 \cos (t)-e^6 (\cos (3-t)-2 \sin (3-t))+5 \sin (t)\right )+1 & \text {True} \\ \\ \\ \\ \\ \end{align*}