## Friday Review

Here are a few problems to review from Unit 2. We discussed the ERF, undetermined coefficients, and variation of parameter, mainly focusing on the ERF.

1. $y''+7y'+12y=3\cos(2x)$
2. $\ddot{x}+x=3\sin(2t)+3\cos(2t)$
3. $\ddot{x}+\dot{x}+2x=5\sin(2t)+\cos(2t)$

From Unit 1 we discussed:

versus

where $t_0$ is the time the object reaches its maximum height (when $v=0$) and upward is positive.

Last, I introduced the Laplace transform:

For example

and finally (this one is miraculous):

### 2 Responses to “Friday Review”

1. gary says:

Wow. That last formula is interesting. So if $f(t)=t$ then $f'(t)=1$ so

OK. $\mathop{\mathscr{L}}\left\{\frac{d}{dt}[t]\right\}=\mathop{\mathscr{L}}\{1\}=\frac{1}{s}$ and $f(0)=0$, so

2. andrew says:

What is f hat? And what is s?

-Andrew