If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Worked example: Euler's method

AP.CALC:
FUN‑7 (EU)
,
FUN‑7.C (LO)
,
FUN‑7.C.4 (EK)

## Video transcript

now that we are familiar with Euler's method let's do an exercise the tests are mathematical understanding of it or at least the the process of using it so this is considered the differential equation the derivative of Y with respect to X is equal to 3x minus 2 I let Y is equal to G of X be a solution to the differential equation with the initial condition G of 0 is equal to K where K is constant Oilers method starting at x equals 0 with the step size of 1 gives the approximation the G of 2 is approximately 4.5 find the value of K so once again this is saying hey look we're going to start with this initial condition when X is equal to 0 Y is equal to K we're going to we're going to use Euler's method with the step size of 1 so we're essentially going to use we're going to step once from 0 to 1 and then again from 1 to 2 and then that approximation is going to give us 4.5 and so given that we started at K we should be able to figure out what K was to get us to G of 2 being approximated as 4.5 so with that I encourage you to pause the video and try to figure this out on your own I am assuming you have tried to figure this out on your own now we can do it together and I'll do the same thing that we did in the first video on Euler's method I'll make a little table here so let me make a little table I can draw a straighter line than that that's only marginally straighter but it'll get the job done so let's make this column X let's I'm going to give myself some space for Y I'm going to do some calculation here Y and then a dy dy DX now we can start at our initial condition when X is equal 0 Y is equal to K when X is equal to 0 Y is equal to K and so what's our derivative going to be at that point well dy/dx is equal to 3x minus 2y so in this case it's 3 times 0 minus 2 times K which is just equal to negative 2 K and so now we can increment one more step we have a step size different color we have a step size of one so we're going to each in each step we're going to increment in each step we're going to increment X by one and so we're now going to be at one now what's our new y going to be well if we increment X by one and our slope is negative two K that means we're going to increment Y by negative two K times one or just negative two K so negative two K so k plus negative two K is negative K so our approximation using Euler's method gets us the point 1 negative K and then what is going to be our slope starting at that point so one negative K our slope is going to be three times our X which is one minus two times our Y which is negative K now and this is equal to three plus two K 3 plus two K and now we'll do another step of 1 because that's our step size another whoops I'm going to get to two and this is the one that we care about right because we're trying to approximate G of two so we have to say well what is our approximate what is our approximation give us for y when x is equal to two and if we're going to have something expressed in K but they're saying that's going to be four point five and then so we could use that to solve for K so what's this going to be so if we increment by one in X we should increment our Y by one times three plus two K which is just going to be so we're going to increment by three plus two K 3 plus two K or negative K plus three plus two K is just going to be three plus K 3 plus K and they're telling us that our approximation gets that to be four point five so three plus K is equal to four point five so the K that we started with must have been if we just subtract three from both sides this is a decimal here it must have been K must be equal to one point five and you can verify that if this initial condition right over here if G of zero is equal to one point five G of zero is equal to one point five and you put 1.5 over here then over here you would get 4.5 and we're done
AP® is a registered trademark of the College Board, which has not reviewed this resource.