# How to solve this problem using laplace transform?

1. Nov 17, 2015

### haha1234

1. The problem statement, all variables and given/known data
The differential equation given:
y''-y'-2y=4t2

2. Relevant equations

3. The attempt at a solution
I used the laplace transform table to construct this equation,and then I did partial fraction for finding the inverse laplace transform.But I'm now stuck at finding the inverse laplace transform of 1/s^3 and 1/s^2...

And the attached photo is the attempted solution.
View attachment 92000

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2. Nov 17, 2015

### Staff: Mentor

I didn't verify your work, but here is a table of Laplace transforms - http://web.stanford.edu/~boyd/ee102/laplace-table.pdf

3. Nov 18, 2015

### haha1234

4. Nov 18, 2015

### Staff: Mentor

Look just below the one for 1/s2. It's a more general formula.

5. Nov 18, 2015

### Ray Vickson

If you know the inverse transform of 1/s, then you can get the inverse transform of 1/s^2 by integration, and of 1/s^3 by integration again. Remember: there are some standard general transform results that are helpful. Below, let $f(t) \leftrightarrow g(s) = {\cal L}(f)(s)$. Then:
$$\begin{array}{l} \displaystyle \frac{df(t)}{dt} \leftrightarrow s g(s) - f(0+)\\ \int_0^t f(\tau) \, d \tau \leftrightarrow \displaystyle \frac{1}{s} g(s) \end{array}$$
These were given specifically in the table suggested by Mark44; did you miss them?