# Laplace transforms to solve initial value DE / partial fractions

• CrazyCamo
In summary, the conversation is about a physics problem involving Laplace transforms. The problem is to solve an initial value problem using Laplace transforms and to sketch the solution over a specific range. The person asking for help has gotten stuck and is seeking guidance on how to proceed. They have attempted to break down the problem but are struggling with the partial fractions.

#### CrazyCamo

Hey guys, i have read many posts on physics forums but this would be my first post. I am quite stuck so any help would be much appreciated.

## Homework Statement

Use Laplace transforms to solve the initial value problem:

f''(y) + 4f'(y) +8y = u(t-1) where y(0) = 1 and y'(0) = -1

Solve this problem using laplace transforms, showing all steps in your reasoning. State the solution y(t) for each of 0<t<1 and t>1, then sketch it over the range 0<= t <= 10, noting its main features.

## The Attempt at a Solution

I have gotten up to F(Y) = (e^-s + s^2 + 3s)/(s(s^2 + 4s+8))

However, from here i am not sure what to do. I tried taking the partial fractions of:

1/(s(s^2 + 4s+8))

but am getting very confused. Again any help would be much appreciated. Cheers

Why don't you break F(Y) down into the sum of its various components?

E.g., D = (s(s^2+4s+8))

F(Y) = (e^-s)/D + s^2/D + 3s/D

You can tackle each term individually.

PS: finding the PFE of 1/D doesn't help.

The denominator is $$s(s^2+4s+ 8)= s(s^2+ 4s+ 4+ 4)= s((s+ 2)^2+ 4)$$ so you can use "partial fractions to write that as $$\frac{A}s+ \frac{Bs+ C}{(s+2)^2+ 4}$$.