IVP with Laplace Transform

Question

My attempt:

Y = Laplace

$$s^2Y -sy(0) - y'(0) - 3Y = ??$$

How do I set up $$h(t)$$ in the form of laplace?

Answer 1

There are some rules/standard results that will help you here. But if you don't know them, why not just explicitly calculate the Laplace transform of $h$?

$$L[h](s) = \int_0^\infty e^{-st} h(t) \ dt = \int_0^2 \frac{t}{2} e^{-st} \ dt + \int_2^\infty e^{-st} \ dt$$

$$= \ ...$$

Answer 2

In addition to the other answer, you can see my answer of Using laplace transforms to solve a piecewise defined function initial value problem.

We can write $h(t)$ using the Heaviside Step Function as:

$$h(t) = \dfrac{t}{2} -\dfrac{t}{2}~ u(t-2) + u(t-2)$$

The Laplace Transform of this is:

$$\mathscr{L} (h(t)) = \mathscr{L} \left(\dfrac{t}{2} -\dfrac{t}{2}~ u(t-2) + u(t-2)\right) = \dfrac{1-e^{-2 s}}{2 s^2}$$

The Laplace Transform of the other part with initial conditions yields:

• $\mathscr{L} (y''(t)) = s^2y(s) -s y(0) -y'(0) = s^2y(s)-1$
• $\mathscr{L} (3y(t)) = 3y(s)$

I will assume you can take it from here.

Answer 3

This is what I have so far.. Am I on the right track? I'm stuck on converting the expressiong with $$e^-2s$$