Solving DE involving dirac delta function

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September 4th 2008, 09:18 PM
dgmossman

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Solving DE involving dirac delta function

I am trying to solve find a solution to a). i can use maple to solve it, but does anyone know how. The "sum of" part is confusing me. Thanks in advance for any help.
September 4th 2008, 11:05 PM
Matt Westwood

The integral of $\delta \left({f \left({x}\right)}\right)$ is just $f \left({x}\right)$ if I remember it correctly ...

So see which instances of the delta function are included within the limits of the integral and just add up the occurrences of f(x) at those instances.
September 5th 2008, 01:08 AM
mr fantastic

Quote:

Originally Posted by dgmossman

I am trying to solve find a solution to a). i can use maple to solve it, but does anyone know how. The "sum of" part is confusing me. Thanks in advance for any help.

I'd suggest using solving using the method of Laplace transforms.

@Matt: I'm not so sure about the result you state. My memory is that $\delta (g(x)) = \sum_{i} \frac{\delta (x - x_i)}{g'(x_i)}$ where the $x_i$ are the real roots of g(x). Therefore the integral is $\sum_{i} \frac{1}{g'(x_i)}$ .
September 5th 2008, 10:42 AM
Matt Westwood

Oh yeah, I knew it was something like that, I couldn't remember for certain - apologies.
September 7th 2008, 08:55 PM
dgmossman

can someone please help me with finding the laplace transform of f(t) (shown in my first post)
September 7th 2008, 10:38 PM
mr fantastic

Quote:

Originally Posted by dgmossman

can someone please help me with finding the laplace transform of f(t) (shown in my first post)

You should be familiar with the Dirac delta function and its properties, in particular the sifting property: $\int_{-\infty}^{+\infty} \delta (t - a) \, f(t) \, dt = f(a)$ .

You should also be familiar withy the definition of a Laplace transform: $LT[f(t)] = \int_{0}^{+\infty} e^{-st} f(t) \, dt$ .

Using the above you should be able to show that $LT[\delta (t - a)] = e^{-as}$ .

Alternatively you can always refer to a standard table of Laplace transforms.

Therefore the Laplace transform of your function is $\sum_{k=0}^{K} \left(e^{-vs/d}\right)^k$ .

Use the formula for the sum of a geometric series to evaluate this sum.