Internal problem ID [4959]
Book: ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG,
EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section: Chapter 6. Laplace Transforms. Problem set 6.4, page 230
Problem number: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+5 y^{\prime }+6 y-\theta \left (t -1\right )-\left (\delta \left (t -2\right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 60
dsolve([diff(y(t),t$2)+5*diff(y(t),t)+6*y(t)=Heaviside(t-1)+Dirac(t-2),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
\[ y \relax (t ) = \frac {\left (2 \,{\mathrm e}^{-3 t +3}-3 \,{\mathrm e}^{-2 t +2}+1\right ) \theta \left (t -1\right )}{6}+\frac {\left (-6 \,{\mathrm e}^{-3 t +6}+6 \,{\mathrm e}^{-2 t +4}\right ) \theta \left (t -2\right )}{6}-{\mathrm e}^{-3 t}+{\mathrm e}^{-2 t} \]
✓ Solution by Mathematica
Time used: 0.244 (sec). Leaf size: 71
DSolve[{y''[t]+5*y'[t]+6*y[t]==UnitStep[t-1]+DiracDelta[t-2],{y[0]==0,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} e^{-3 t} \left (-1+e^t\right ) & t\leq 1 \\ \frac {1}{6} e^{-3 t} \left (6 e^4 \left (-e^2+e^t\right ) \theta (t-2)+e^{3 t}-3 e^t \left (-2+e^2\right )+2 e^3-6\right ) & \text {True} \\ \\ \\ \\ \\ \end{align*}