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The method is the following and it's an altered version of Euler's method:

$$y_{n+1}=y_n+hf(x_n+h/2,y_n+hy'_n/2) $$

How can I prove its order of convergence? All I can get from my textbook is how I can prove Euler's method order. But the way it's done is starting from the Taylor series and its next term. But I can't do this here can I? Also, I don't quite get this method and what is going on in the parenthesis. How did this method come up?

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    This is also called the explicit midpoint method. You need to show a local error of order 3 to conclude to a global error of order 2. And yes, this can be relatively easily (compared to 3rd and 4th order methods) done via Taylor expansion. – 2017-02-10
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    "But I can't do this here can I?": why not ? – 2017-02-10
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    How can I show the local error of order 3? And how can I use the taylor expansion and know what term to stop at? – 2017-02-10
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    It of course depends on how you define the local truncation error. I was using $e(h)=y(x_n+h)-y(x_n)-h\Phi_f(x_n,y(x_n),h)$ with an exact solution of $y'=f(x,y)$. Computing its derivatives, always replacing $y'(x)$ with $f(x,y(x))$, and how they evaluate at $h=0$ will tell you the order as the first coefficient that is not zero. – 2017-02-10

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