Question

Find the inverse Laplace transform of F(s)=2/(s^2+3s-4).

Answer 1

So looks like we need to factor the denominator, and then apply Partial Fraction Decomposition.

Are you stuck on any particular part? :)

Answer 2

After factoring the denominator, I don't know what to do. Can you help me step by step?

Answer 3

Partial Fraction Decomposition tells us that the factors in the denominator can be broken down like this ~ Where A and B are unknown costants.
\[\large \frac{2}{(s-1)(s+4)} \qquad=\qquad \frac{A}{s-1}+\frac{B}{s+4}\]

From here, we'll multiply both sides by the denominator on the left giving us,\[\large 2=A(s+4)+B(s-1)\]

Confused by anything yet?

Answer 4

What's next?

Answer 5

We'll plug in specific values for \(\large s\) in order to solve for A and B.
We want values that will zero one of the unknowns.

\[\large s=1 \qquad\to\qquad 2=A(1+4)+B(1-1) \qquad\to\qquad 2=5A+0\]So that get's us our A value.
To get B we use the other convenient value for s.

\[\large s=-4 \qquad\to\qquad 2=A(-4+4)+B(-4-1) \qquad\to\qquad 2=-5B\]

Answer 6

Wait a minute, please don't leave.

Answer 7

k :)

Answer 8

And then?

Answer 9

So we've found our A and B.
\[\large A=\frac25 \qquad\qquad \qquad B=-\frac25\]
Using our initial set up,\[\large \frac{2}{(s-1)(s+4)} \qquad=\qquad \frac{A}{s-1}+\frac{B}{s+4}\]
That tells us that our fraction can be written like this,\[\large \frac{2}{(s-1)(s+4)} \qquad=\qquad \frac{\frac25}{s-1}-\frac{\frac25}{s+4}\]

Answer 10

Let's pull those ugly fractions out of the numerator so it's a little easier to read.\[\large \frac{2}{5}\left(\frac{1}{s-1}\right)-\frac{2}{5}\left(\frac{1}{s+4}\right)\]

Answer 11

These will be nice and easy to take the Inverse Laplace of now.
Just need to remember back to your rules.\[\large \mathscr{L}\left[e^{at}\right] \qquad=\qquad \frac{1}{s-a}\]
So let's apply this to our first term but in reverse,\[\large \mathscr{L}^{-1}\left[\frac{2}{5}\frac{1}{s-1}\right] \qquad=\qquad \frac{2}{5}\mathscr{L}^{-1}\left[\frac{1}{s-1}\right]\]

Answer 12

Understand how to take the inverse of that first term there?

Answer 13

Thank you so much for the help.

Answer 14

No prob c:
Partial Fractions can be a bit tough if you don't remember them.
You might wanna do a few more of these to brush up on them.