Internal problem ID [11535]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 3, Laplace transform. Section 3.4 Impulsive sources. Exercises page
173
Problem number: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }-x=\delta \left (t -5\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 25
dsolve([diff(x(t),t$2)-x(t)=Dirac(t-5),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
\[ x \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -5\right ) \left (-{\mathrm e}^{-5+2 t}+{\mathrm e}^{5}\right ) {\mathrm e}^{-t}}{2} \]
✓ Solution by Mathematica
Time used: 0.058 (sec). Leaf size: 31
DSolve[{x''[t]-x[t]==DiracDelta[t-5],{x[0]==0,x'[0]==0}},x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {1}{2} e^{-t-5} \left (e^{2 t}-e^{10}\right ) \theta (t-5) \]