Internal problem ID [864]
Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima,
Meade
Section: Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number: 10(a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{2}+y=\delta \left (-1+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.562 (sec). Leaf size: 28
dsolve([diff(y(t),t$2)+1/2*diff(y(t),t)+y(t)=Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \frac {4 \,{\mathrm e}^{\frac {1}{4}-\frac {t}{4}} \operatorname {Heaviside}\left (t -1\right ) \sqrt {15}\, \sin \left (\frac {\sqrt {15}\, \left (t -1\right )}{4}\right )}{15} \]
✓ Solution by Mathematica
Time used: 0.097 (sec). Leaf size: 40
DSolve[{y''[t]+1/2*y'[t]+y[t]==DiracDelta[t-1],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {4 e^{\frac {1}{4}-\frac {t}{4}} \theta (t-1) \sin \left (\frac {1}{4} \sqrt {15} (t-1)\right )}{\sqrt {15}} \]