Internal problem ID [861]
Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima,
Meade
Section: Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+4 y-2 \left (\delta \left (t -\frac {\pi }{4}\right )\right )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 16
dsolve([diff(y(t),t$2)+4*y(t)=2*Dirac(t-Pi/4),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = -\theta \left (t -\frac {\pi }{4}\right ) \cos \left (2 t \right ) \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 25
DSolve[{y''[t]+4*y[t]==2*DiracDelta[t-Pi/4],{y[0]==0,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \sin (t) \cos (t)-\theta (4 t-\pi ) \cos (2 t) \\ \end{align*}