Internal problem ID [11539]
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: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y^{\prime }+y=\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 28
dsolve([diff(y(t),t$2)+diff(y(t),t)+y(t)=Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \frac {2 \sqrt {3}\, \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{\frac {1}{2}-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, \left (t -1\right )}{2}\right )}{3} \]
✓ Solution by Mathematica
Time used: 0.124 (sec). Leaf size: 40
DSolve[{y''[t]+y'[t]+y[t]==DiracDelta[t-1],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {2 e^{\frac {1}{2}-\frac {t}{2}} \theta (t-1) \sin \left (\frac {1}{2} \sqrt {3} (t-1)\right )}{\sqrt {3}} \]