Here's an overview of Euler's method

for solving differential equations of this form.

The derivative of some unknown function X is just a function of X,

and we assume that we know the starting value of this function X,

- so this could be temperature, this could be position, could be anything.

So, we start - we need to choose a step size,

- that's a choice that we get to make, and we'll start at time t equal 0,

where we know the X value.

So, then using the current value of X,

the equation - the differential equation - this function,

tells us the rate of change.

So, we know how fast X is changing.

We then use this rate of change

to determine the next value for X,

This is the current value for X,

this is how much X changes, in the time interval delta t.

This is a little bit of make-believe

- we're pretending that the rate of change is constant

for this time interval delta t, and we can use that

to figure out the next value of X,

then, we increased t by delta t

and we go back to this step, and we figure out the derivative again.

The derivative tells us how to go forward a little bit in time

to figure out the next X, then we update time, figure out the derivative again,

move forward to figure out X.

So 2 and 3 - these are the key steps here,

- wer'e constantly shuffling back and forth,

the function - the differential equation - tells us the derivative,

- here we use the derivative to figure out the value of X,

and then we go back, X gives us the derivative from the differential equation,

- we use a derivative to figure out X, and so on.

So, one repeats through these processes

until you have enough of a solution.

So, in practice, one would choose smaller and smaller delta t

until the solution curve stops changing.

So, if you chose a delta t of 2, and then 1, and then .01, and .001,

doing this on a computer, or a spreadsheet,

and eventually one would see that your table of values, if you plotted them,

your X's would stop changing, and that would be an indication

that you're delta t was small enough.

So, this is Euler's method in a nutshell.

It gets at the heart of what the differential equation means,

- the rate of change is given by the quantity X,

we use that rate of change to figure out X,

so, again we're thinking of this as a dynamical system,

- it's a rule that specifies how X changes,

the rule is written in terms of the derivative,

- the rate of change of X, rather than X directly,

but, not a problem, Euler's method, or things like it,

let us convert from rates of change in to the function itself.