# D.E. - Piecewise Laplace Transforms

Greetings all/any:

## Homework Statement

Take the Laplace transform of the following initial value problem and solve for Y(s) = L{y(t)}
y'' - 3y' -28y = 1 if 0<t<1
0 if 1<t

initial conditions: y'(0) = y(0) = 0

a. Y(s) = ?
b. now find the inverse transform to find y(t). Use step(t-c) for uc(t)

*Note:
1/(s(s-7)(s+4)) = (-1/28)/s + (1/44)/(s+4) + (1/77)/(s-7)

## Homework Equations

not entirely sure. I know random equations for the left hand side of the shifted equation, but I'm lost as to how to even start applying them. Nevertheless:

L{f(t-a) u(t-a)} = F(s)e-as

L{g(t) - g(t-a)f(t-a)}

...etc? again, I have all of these equations, but I don't know how to use them so even if I found out some way to just plug numbers in, I still wouldn't understand what the concept behind this is.

## The Attempt at a Solution

I can solve the left hand side:

y'' - 3y' -28y = f(t); y(0)=y'(0)=0

-3L{y'(t)} = sL{y(t)} - y(0) = -3(sY(s) - 0) = -3sY(s)

L{y''(t)} = s2L{y(t)} -sy(0) -s'(0) = s2Y(s) - 0s - 0 = s2Y(s)

= [s2Y(s)] -3[sY(s)] - 28[Y(s)] = F(s)

= Y(s) [s2 -3s -28]= F(s)

....and then I'd have to find the laplase tranform of the right hand side as well, and divide through to isolate and solve for Y(s)

That said. I have no idea where to start on the right hand side. I've looked at the book, (Edwards & Penny 4th ed.), looked at notes, tried finding stuff online - basically what I get is a lot of examples of solutions, but I don't understand what they're doing to find their right hand side equations...

Thank you.

## Answers and Replies

Related Calculus and Beyond Homework Help News on Phys.org
Cyosis
Homework Helper
Your right hand side is 1. What is the Laplace transform of 1?

Defennder
Homework Helper
You can interpret the RHS of 1 to be the unit step function u(t) if you're using a one-sided Laplace transform.