Internal problem ID [11536]
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: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+x=\delta \left (-2+t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 13
dsolve([diff(x(t),t$2)+x(t)=Dirac(t-2),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
\[ x \left (t \right ) = \operatorname {Heaviside}\left (t -2\right ) \sin \left (t -2\right ) \]
✓ Solution by Mathematica
Time used: 0.206 (sec). Leaf size: 17
DSolve[{x''[t]+x[t]==DiracDelta[t-2],{x[0]==0,x'[0]==0}},x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to -\theta (t-2) \sin (2-t) \]