MATH 20D Lecture 25

Example. Use Laplace transform to solve the system

Apply Laplace Transform to both equations

augmented matrix

Multiply 2nd equation by then subtract the first

So

Now we find inverse Laplace transform of and

Bottom =

Let’s inverse

So and


You try to inverse to get

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: population of fish in a lake at time (in years)

Simplest logistic Equation: and (some constant)

Suppose we want to restock the fish at a constant rate of unit of fish per year.
New equation: .
We can easily solve this new equation.

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What if we run out of money and restocking program ends after just 6 months?

where

What if during the 2nd half of the year, we fundraise enough to restart restocking fish and we can restock the fish at rate (unit of fish per year) for another 4 months

where

7.6 - Laplace Transform of Unit Step Functions

Heaviside/Unit step function

The heaviside step function or unit step function, denoted is a discontinuous function defined by

Example. Let $c>0$. Express the following function in terms of the unit step functions

Example. Express the following in terms of the unit step functions:

When :

When :
but
When :
so

Theorem.

For


We make the substitution so when and .
Thus,