Laplace transform of Dirac delta

Question

I need some assistance with a question involving the Dirac delta

$$y''(t)=t^3δ(t-2), y(0)=y'(0)=0$$

I'm assuming the first step is

$$s^2Y(s)-sy(0)-y'(0)=L((t^3)δ(t-2))$$

but I'm having trouble finding the Laplace of the RHS

any advice would be appreciated thanks

Answer 1

Integrals with Dirac functions are so easy that they're hard. $\delta(t-2)=0$ everywhere except when $t=2$, and then it has weight 1. So $\int_0^{\infty} \delta(t-2)f(t) \; dt = f(2).$