How can I use the Dirac delta function in a Laplace transform?

[Jamin2112]

Homework Statement

2y'' + y' + 4y = ∂(t-π/6)sin(t); y(0)=0, y'(0)=1/2.

Homework Equations

Dunno

The Attempt at a Solution

The left side of the equation is what's tripping me up. There's nothing on my Elementary Laplace Transforms table that has the Dirac delta function multiplied by another function, or sin(at) multiplied by another function f(t). So what should I do?

 

[Cyosis]

You mean the right side? The delta functions makes it very easy actually. Consider the value of the integral when $t \neq \pi/6$. Now consider $t = \pi/6$.

 

[vela]

I think you meant the right-hand side. Anyway, just plug the RHS into the definition of the Laplace transform and evaluate the integral. The delta function makes it easy to do.

 

[gabbagabbahey]

One of the properties of the Dirac delta is that for any sufficiently smooth function $f$, $\int f(x)\delta(x-a)dx=f(a)$ if the integration interval includes the point $x=a$, and the integral is zero otherwise. Use that.