Internal problem ID [4957]
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]]
\[ \boxed {y^{\prime \prime }+5 y^{\prime }+6 y-\left (\delta \left (t -\frac {\pi }{2}\right )\right )-\cos \left (t \right ) \operatorname {Heaviside}\left (-\pi +t \right )=0} \] 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: 79
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)
\[ y \left (t \right ) = -\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-3 t +\frac {3 \pi }{2}}-\frac {3 \operatorname {Heaviside}\left (-\pi +t \right ) {\mathrm e}^{-3 t +3 \pi }}{10}+\frac {2 \operatorname {Heaviside}\left (-\pi +t \right ) {\mathrm e}^{-2 t +2 \pi }}{5}+\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-2 t +\pi }+\frac {\operatorname {Heaviside}\left (-\pi +t \right ) \left (\cos \left (t \right )+\sin \left (t \right )\right )}{10} \]
✓ Solution by Mathematica
Time used: 0.375 (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*}