A smaller delta t makes Euler's method

more accurate, and we can see why:

The reason Euler's method isn't accurate

is that we're pretending that a constantly changing rate

is actually constant over some time interval.

Over a time interval of 2 minutes, that rate might change quite a bit,

but the rate will change less if the time interval is smaller.

So, if the time interval is 1 minute, instead of 2,

the bit of make-believe, where we pretend that

the rate isn't changing, will be closer to reality,

and I can illustrate that on this next plot.

So, I won't go through all the numerics of this,

but here is Euler's method

for 2 different delta t's.

So, first, the squares, which we've already seen,

we calculated those before,

that's Euler's method with a delta t of 2,

where we're pretending that a constantly changing rate

is actually constant for a whole two minutes.

Delta T of 1, that's the triangles with the dash, and not dotted line.

a little hard to see, but the key thing is it's between these two

- it's closer to the exact solution which is the solid curve.

It's closer because the ignoring of the problem is a less bad thing to do,

- we're pretending now that a continually changing rate

is only constant for 1 minute, instead of 2, so it's not as much of a lie,

and by now you can probably guess

how we could make this better and better and better

- we would let delta t get smaller and smaller and smaller,

and then we would see that the Euler method

would be exactly on top of this line.

Now that we've seen part of a particular example

let me talk about Euler's method a bit more generally.

Euler's method applies to differential equations of this form.

A differential equation is a dynamical system,

a rule for how something changes in time.

What makes differential equations a little bit tricky

is that the rule is indirect.

This tells us how the derivative changes

and we're interested in how the quantity X itself changes.

Euler's method is just a way to go from this

indirect information derivative, to the direct information about X

So Euler's method converts this indirect rule,

the differential equation - indirect rule involving the derivative

- the rate of change, and it converts that in to values for X.

It does so, by pretending that this rate of change

is constant over a time interval.

So, Euler's method does this conversion by pretending that the derivative,

which is constantly changing, is actually constant

over some time interval delta t.

This bit of make-believe gets better

closer to the true value, as delta t gets smaller.

So, as delta t, our time interval over which were pretending the rate isn't changing,

as delta t gets closer and closer to 0,

this Euler pretending will get less and less wrong,

and in this way, Euler's method, a solution obtained

from Euler's method, will get closer and closer to the true answer.

So, as delta t gets closer and closer to 0,

a solution obtained by Euler's method will get

closer and closer to the exact solution.

So, Euler's method is a computational way of

finding a solution to a differential equation.

It requires doing a computation,

and you can see that as delta t gets smaller and smaller

the computational will get longer and longer.

- we'll need to do more and more steps to get anywhere,

so these are almost always done on a computer.

So, this is an algorithmic solution to differential equations.

It's a procedure, it's well-defined,

for well-defined differential equations, it's guaranteed

to converge to the exact solution.

So, Euler's method is very general,

it almost always works, and I think it gets at

the core idea of a differential equation:

a differential equation is a dynamical system,

a rule for how something changes.

The rule is a little bit indirect, because it is in terms of the derivative,

the rate of change of this quantity X, and not X itself

but Euler's method is a little bit of a trick

that converts this indirect information about the derivative

into direct information about values for X