Internal problem ID [12913]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 6. Laplace transform. Section 6.4. page 608
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+2 y=-2 \left (\delta \left (-2+t \right )\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.219 (sec). Leaf size: 32
dsolve([diff(y(t),t$2)+2*diff(y(t),t)+2*y(t)=-2*Dirac(t-2),y(0) = 2, D(y)(0) = 0],y(t), singsol=all)
\[ y = -2 \operatorname {Heaviside}\left (-2+t \right ) {\mathrm e}^{2-t} \sin \left (-2+t \right )+2 \,{\mathrm e}^{-t} \left (\cos \left (t \right )+\sin \left (t \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.3 (sec). Leaf size: 31
DSolve[{y''[t]+2*y'[t]+2*y[t]==-2*DiracDelta[t-2],{y[0]==2,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to 2 e^{-t} \left (e^2 \theta (t-2) \sin (2-t)+\sin (t)+\cos (t)\right ) \]