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## 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 […]

## Fourier Series

Let $f$ be a piecewise continuous function on the interval $[-T,T]$.  The Fourier Series of $f$ is the trigonometric series

where the $a_n$'s and $b_n$'s are given by the formulas

For example, let $f(x)$ be defined by !f(x)=\begin{cases} 0 & , […]

## Bernoulli Equations

Bernoulli equations are of the form:

Dividing by $y^n$:

If $n=0$ or $n=1$ then the equation is linear.  Otherwise a substitution of $u=y^{1-n}$ will give us a linear equation. If $u=y^{1-n}$, then $\frac{du}{dx}u=(1-n)\frac{dy}{dx}y^{-n}$.  Substituting these into the second equation from above:

For example,

Dividing by $y^4$, we get  $y'y^{-4}+2xy^{-3}=x$ Substituting $u=y^{-3}$ […]

## Homogeneous First-order Equations

There are two ways to view "homogeneous" first-order ordinary differential equations. 1. $\frac{dy}{dx}=f(x,y)$ is a general first order ordinary differential equation.  If the function on the right, $f(x,y)$, can be rewritten as a function of $\frac{y}{x}$, e.g. $f(x,y)=\frac{y-2x}{x}$ can be rewritten as $g\left(\frac{y}{x}\right)=\frac{y}{x}-2$.  If we let $v=\frac{y}{x}$, then we would have $g(v)=v-2$. In general, the […]