Internal problem ID [10994]
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 5(e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {4 y^{\prime \prime }+4 y^{\prime }+y-{\mathrm e}^{-\frac {t}{2}} \left (\delta \left (-1+t \right )\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 17
dsolve([4*diff(y(t),t$2)+4*diff(y(t),t)+y(t)=exp(-t/2)*Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -1\right ) \left (t -1\right ) {\mathrm e}^{-\frac {t}{2}}}{4} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 23
DSolve[{4*y''[t]+4*y'[t]+y[t]==Exp[-t/2]*DiracDelta[t-1],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{4} e^{-t/2} (t-1) \theta (t-1) \\ \end{align*}