Internal problem ID [4958]
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: 10.
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-\left (\delta \left (t -\frac {\pi }{2}\right )\right )-\theta \left (-\pi +t \right ) \cos \relax (t )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 62
dsolve([diff(y(t),t$2)+5*diff(y(t),t)+6*y(t)=Dirac(t-1/2*Pi)+Heaviside(t-Pi)*cos(t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = \frac {\left (\cos \relax (t )+\sin \relax (t )-3 \,{\mathrm e}^{-3 t +3 \pi }+4 \,{\mathrm e}^{-2 t +2 \pi }\right ) \theta \left (-\pi +t \right )}{10}+\theta \left (t -\frac {\pi }{2}\right ) \left (-{\mathrm e}^{-3 t +\frac {3 \pi }{2}}+{\mathrm e}^{-2 t +\pi }\right ) \]
✓ Solution by Mathematica
Time used: 0.61 (sec). Leaf size: 79
DSolve[{y''[t]+5*y'[t]+6*y[t]==DiracDelta[t-1/2*Pi]+UnitStep[t-Pi]*Cos[t],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{10} e^{-3 t} \left ((\theta (\pi -t)-1) \left (-4 e^{t+2 \pi }-e^{3 t} (\sin (t)+\cos (t))+3 e^{3 \pi }\right )-10 e^{\pi } \left (e^{\pi /2}-e^t\right ) \theta (2 t-\pi )\right ) \\ \end{align*}