Internal problem ID [2880]
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]]
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+5 y=5 \operatorname {Heaviside}\left (t -3\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 4.297 (sec). Leaf size: 53
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)
\[ y \left (t \right ) = \left (-\frac {1}{2}-i\right ) \operatorname {Heaviside}\left (-3+t \right ) {\mathrm e}^{\left (-2-i\right ) \left (-3+t \right )}+\left (-\frac {1}{2}+i\right ) \operatorname {Heaviside}\left (-3+t \right ) {\mathrm e}^{\left (-2+i\right ) \left (-3+t \right )}+\operatorname {Heaviside}\left (-3+t \right )+{\mathrm e}^{-2 t} \left (2 \cos \left (t \right )+5 \sin \left (t \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 68
DSolve[{y''[t]+4*y'[t]+5*y[t]==5*UnitStep[t-3],{y[0]==2,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-2 t} (2 \cos (t)+5 \sin (t)) & t\leq 3 \\ e^{-2 t} \left (-e^6 \cos (3-t)+e^{2 t}+2 \cos (t)+2 e^6 \sin (3-t)+5 \sin (t)\right ) & \text {True} \\ \end {array} \\ \end {array} \]