Laplace transform of Dirac delta

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

• What's the definition of the Laplace transform? What do Dirac delta functions do to integrals? – Sean Lake Sep 12 '16 at 10:09
• The RHS after your first step is no correct. You should write $L\left(t^{3}\delta(t-2)\right)$ and follow the @SeanLake's hints. – Alex Silva Sep 12 '16 at 10:15
• I forgot about the brackets but thats what i was meant to type, its fixed now – MathNoob Sep 12 '16 at 10:26

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 $\int_0^{\infty} \delta(t-2)f(t) \; dt = f(2).$