In this segment we'll talk about the Euler’s method of solving
ordinary differential equations. We'll concentrate on the theory aspect of it.
So, Euler’s method works for solving first order ordinary differential equations
of the form as follows. So, what that means is that it has to be a first order ordinary
differential equation. It has to be written in the form that the slope is of the dependent
variable with respect to the independent variable is given as a function of x comma y
and then of course you have to have some initial condition so as to be able to solve for y
at other points. Now this can be a little bit challenging to a student who has seen
this for the first time because they're generally familiar with the regular format
So, let's take an example you might be given a differential equation like this
from your ordinary differential equations course,
and you might say that hey how do I write it in this particular format right here all you have
to do is to just rewrite it you'll say three dy by dx is equal to this quantity. Right,
and then you will say dy by dx is equal to this quantity
and this becomes your function of x comma y. Along with the condition that you need which
is y sub 0 to be 3 in this case and that's about it. So rewriting your differential equation as
generally given into this format is pretty straightforward and the reason why you do that is
because what Euler’s method is based on or many of the other methods are based on is that if you know
the approximate value of y at a particular point and an approximate value of the derivative of the
dependent variable respect to the independent variable first slope first derivative at that
particular point then you can approximately find the value of y at some other point.
Let's do this graphically first. The whole point about solving differential equation,
which is of this form let's suppose, is that you want to find y as a function of x. That's the
ultimate goal of solving an ordinary differential equation. The only thing which will be different
from your ODE class would be that will you find the value of y at discrete points as opposed to
as a continuous function. So, what we're going to do is we are going to say hey let's suppose I know
that this is let's suppose x0. If this is x0, I know the value of y which will be y0 here because
that is given to me as an initial condition. Then let's suppose the y as a function of x is exactly
known may look like that. But keep in mind that when I’m saying hey it's exactly known I don't
know it because that's why I’m trying to solve the ordinary differential in the first place. but
this is just for illustration purposes. But I know the value of the slope at this particular point.
I know the value of the slope at that particular point because I have written my differential
equation in this format that the slope is given by the function f and so if I draw the slope and then
I take the point x1 that gives me the approximate value of y then at that particular point. So
that'll become x1 comma y1 as my coordinate there and y1 will be the approximate value of y
at x1. Then I can take this, draw the slope here now keep in mind the slope
will be approximate because now I know the value of y only approximately so when I’m going
to substitute in here, I’ll have the exact value of x, but I don't have the exact value of y. But
I will get some approximate value of the slope there and then I can go ahead and say okay hey
this is x2 so that is the y2 value will be the y-coordinate of that and what y2 will represent
is the approximate value of y at x. So that's what the Euler’s method is all about and what
we're going to do is we're going to develop a general formula for it in the next slide.
Let's go and talk about how we can find the general form for the Euler’s method and
given that the format of the ordinary differential equation is given as this
with the initial condition given at x0. So, if we look at it from a graphical point of view,
let's suppose we are at some point x sub i. So, we’ve already gone through several steps now as
shown in the previous slide and let's suppose we’re at the x sub i point. So, we know that at
this particular point we will know some value of y corresponding to x of i. Now if I draw the slope
at this particular point what will happen is that if I then choose the next point x of I plus 1,
this one gives me the approximate value of y at that particular point.
So, the question is that hey, maybe because I know the slope of the function at this particular
point, because that's how the differential equation has been portrayed to me, how will I
find y sub i plus 1? So, if you look at this distance right here, that will be y sub i plus
1 minus y sub i because this one is y sub i plus 1 and this one is y sub i and if you look at this
difference right here that is x sub i plus 1
minus x sub i. So that basically becomes, hey this is the rise, and this is the run. So,
we have rise over run will be equal to the slope of y with respect to x at that point x sub i.
Now what is the rise? We just calculated as y sub i plus 1 minus y sub i. The run is x sub
i plus 1 minus x sub i and the slope of y with respect to x sub i is nothing but the function f
at that particular point which will have the value of x sub i and this approximate value of y sub i.
So, since we have that, that means that the only unknown now here is y sub i plus 1 because
x sub i and x sub i plus 1 is something which we chose. They are independent variables y sub i is
known from the previous step so let's go and calculate what y five plus one would be. y sub
i plus one minus y sub i will be equal to this quantity right here.
And if I want to just write the expression for y sub i plus 1 it will turn out to be this.
So many times, what they do is they call this the step size,
so they call it h and they call it the step size and they rewrite the Euler’s method formula
as y sub i plus 1 equals y sub i plus this quantity right here times h.
And that's what Euler’s method is all about that, hey, if you know the value of the
dependent variable y at a particular point and you know its slope. Slope is the function f
at that particular point some approximate value and you know
the step which you are taking then you can find the value of y at that particular point
And that is the end of this segment.