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I have to do this take home test for school before break ends and it is on Slope Fields and Euler's Method. I'm having trouble understanding and/or working with Euler's Method.

These are the two problems I am stuck on:

2.$\text{}$ Use Euler's Method with $\Delta x=0.1$, $\dfrac{dy}{dx}=2x-y$ and $y=0$ when $x=1$ to find the value of $y$ when $x=1.3$.

A) 0.6      B) 0.2      C) 0.4      D) 0.8

3.$\text{}$ Find the first three approximations (by hand) $y_1$, $y_2$, $y_3$ using Euler's Method for the initial value problem

$$\frac{dy}{dx}=1+y,\quad y(0)=1.$$

(original image)

Thanks for any help or tips in advance!

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Well, do you know what Euler's method is? Where are you stuck? – Tyler Dec 28 '12 at 2:25
@TylerBailey well, my teacher didn't really teach it to us, we just have to figure it out over break. This is AP Calculus BC. So, I've been reading the textbook and and nothing really helps me. None of the example problems are anything similar. – Mitchell Dec 28 '12 at 2:28
I really don't think you should be getting help on a test, even if you could take it home, unless your teacher told you you could. I might be misunderstanding this, in which case I apologise. – user50407 Dec 28 '12 at 2:35
Yeah, she said we can work together, and use any online sources. – Mitchell Dec 28 '12 at 2:36
Ok, then it's alright I suppose. – user50407 Dec 28 '12 at 2:37
up vote 4 down vote accepted

Euler's method in essence is just following the directional field of a differential equation by taking steps through it.

Here we have a simple case of $\frac{\textrm dy}{\textrm dx}=f(x,y)$.

Recall the intuitive idea of the derivative of a function as a function which gives the slope of the tangent to that function. Then by simple linear approximation, starting from a point $(x_0,y_0)$, we get the next point by $x_1=x+\Delta x$, $y_1=y_0+\Delta y\approx y_0+\frac{\textrm dy}{\textrm dx}(x_0,y_0)\Delta x$, where $\Delta x$ is a given in the question. Then we note that $\frac{\textrm d y}{\textrm d x}(x,y)=f(x,y)$, because this is a given.

So to conclude, we start with a point $(x_0,y_0)$, and for every point we get the next point by $x_{n+1}=x_n+\Delta x$ and $y_{n+1}=y_n+f(x_n,y_n)\Delta x$. Simply do this for the given $f(x,y)$ and $\Delta x$ and you should be fine.

I know the point of this exercise is to learn Euler's method, but you should be able to solve this equation without too much difficulty, it is a first order linear differential equation.

I hope this was clear enough, else just post a comment.

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Thank you so much! – Mitchell Dec 28 '12 at 2:53
No problem, but that was a bit fast to have read it all I think. Have a good read of it and decide if you're still as happy with it then :). Like I said just post a comment otherwise. – user50407 Dec 28 '12 at 2:55

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