Displaying posts published in

## Systems recordings

Here is a recoding of a solution to a system of linear, first-order differential equations with a slope field and solution curves near eigenvectors. or get it on You Tube, http://youtu.be/A-RwYm2SLiA Here is a recording of a similar system but with complex eigenvalues: http://youtu.be/q9nCdpdlqE4

## Exam 3

Here is the third test.  I have one more comprehensive part to post: coming soon. For now enjoy part 1 of the last test: mth256_exam3_summer12

## Laplace Transform, impulse response

Suppose a differential equation, $a\ddot{w}+b\dot{w}+cw=\delta(t)$, has the following unit impulse response:

What would its response to the unit step function be?

Using convolution we get

So, $v(t)=\left[-e^{-\tau}+\frac{1}{3}e^{-3\tau}\right]_0^t=-e^{-t}+1+\frac{1}{3}e^{-3t}-\frac{1}{3}=\frac{2}{3}-e^{-t}+\frac{1}{3}e^{-3t}$ Alternatively, we know that $w(t)=\mathop{\mathscr{L^{-1}}}\left\{\frac{1}{as^2+bs+c}\right\}$. Comparing that to $\mathop{\mathscr{L}}\left\{w(t)\right\}=\mathop{\mathscr{L}}\left\{e^{-t}-e^{-3t}\right\}=\frac{1}{s+1}-\frac{1}{s+3}$.

Cool! $a=\frac{1}{2}$ , $b=2$ , and $a=\frac{3}{2}$. For fun we could solve […]

## Laplace Transform Example

Applying the Laplace transform operator to both sides:

And now for the inverse Laplace transform:

The second term is easy:

The first term requires partial fraction decomposition or completing the square and involving complex numbers.  I am going to show the latter.

[…]

## 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. $y''+7y'+12y=3\cos(2x)$ $\ddot{x}+x=3\sin(2t)+3\cos(2t)$ $\ddot{x}+\dot{x}+2x=5\sin(2t)+\cos(2t)$ From Unit 1 we discussed:

versus

where $t_0$ is the time the object […]