Laplace transform on specific interval

[elimenohpee]

Homework Statement

I need to solve the following ODE using the laplace domain:
y' + 4y + 20(integral from 0->t)y(t)dt = 34e^-t on the interval 0<t<4 (y(0) = 0)

Homework Equations

I can solve the ODE fine, no problems. But I'm a little confused on how to evaluate it strictly on this interval. Would I need to utilize the step function somehow?

The Attempt at a Solution

This is what I did so far:
(sY(s) - 0) + 4Y(s) + 20Y(s) / s = 34 / (s + 1)
but this is not taking into account the interval...

 

[mighty2000]

Thank you for your question. To solve this ODE using the Laplace domain, you will indeed need to incorporate the step function. The step function, denoted as u(t), is defined as 0 for t<0 and 1 for t≥0. It essentially represents the "switch" that turns on at t=0.

To incorporate the step function into your Laplace transform, you will need to use the following property:

L{u(t-a)f(t-a)} = e^-asF(s)

In your case, a=0 and f(t) = y(t). So the Laplace transform of y(t) on the interval 0<t<4 would be:

L{y(t)} = e^-0sF(s) = F(s)

Therefore, your Laplace transform equation would become:

(sY(s) - 0) + 4Y(s) + 20Y(s) / s = 34 / (s + 1)

Notice that the integral from 0 to t has been replaced by the Laplace transform of y(t), which is simply F(s).

From here, you can solve for Y(s) and then take the inverse Laplace transform to find the solution for y(t) on the interval 0<t<4. Remember to also incorporate the initial condition of y(0) = 0 into your solution.

I hope this helps. Good luck with your problem!