Internal problem ID [10978]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin.
CRC Press 2015
Section: Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number: Problem 3(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y-\operatorname {Heaviside}\left (t \right )+\operatorname {Heaviside}\left (-1+t \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 41
dsolve([diff(y(t),t$2)+2*diff(y(t),t)+y(t)=Heaviside(t)-Heaviside(t-1),y(0) = 1, D(y)(0) = -1],y(t), singsol=all)
\[ y \left (t \right ) = t \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-t +1}+\left (1+\operatorname {Heaviside}\left (t \right ) \left (-t -1\right )\right ) {\mathrm e}^{-t}+\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 43
DSolve[{y''[t]+2*y'[t]+y[t]==UnitStep[t]-UnitStep[t-1],{y[0]==1,y'[0]==-1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} e^{-t} & t<0 \\ 1-e^{-t} t & 0\leq t\leq 1 \\ (-1+e) e^{-t} t & \text {True} \\ \\ \\ \\ \\ \end{align*}