Internal problem ID [4952]
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: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y-\left (\delta \left (-\pi +t \right )\right )+\delta \left (-2 \pi +t \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 22
dsolve([diff(y(t),t$2)+y(t)=Dirac(t-Pi)-Dirac(t-2*Pi),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
\[ y \left (t \right ) = -\sin \left (t \right ) \left (\operatorname {Heaviside}\left (-\pi +t \right )+\operatorname {Heaviside}\left (-2 \pi +t \right )-1\right ) \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 23
DSolve[{y''[t]+y[t]==DiracDelta[t-Pi]-DiracDelta[t-2*Pi],{y[0]==0,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -((\theta (t-2 \pi )+\theta (t-\pi )-1) \sin (t)) \\ \end{align*}