# Numerical methods: Midpoint rule for integration

#### JhonD

Hi. I want to solve an ode using some numerical integration methods. I have an equation of the form $$\displaystyle y'(t)=f(t,y(t))$$.

Let's say my equation is: $$\displaystyle y'(t)=\mu y(t)+g(t)$$, with $$\displaystyle \mu$$ a constant, and g an arbitrary function.

If I use Euler method, I have that $$\displaystyle y_{n+1}=y_n+hf(t_n,y_n)$$.

So I would have: $$\displaystyle y_{n+1}=y_n+h[\mu y_n+g_n]$$

Now, if I want to use the midpoint rule, I would have:

$$\displaystyle y_{n+1}=y_n+hf(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n,y_n))$$.

The problem I have is with how to intepret the term $$\displaystyle f(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n,y_n))$$, for my example, what would it explicitly be? Would be it ok to take:

$$\displaystyle f(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n,y_n))=\mu y_n+\mu\frac{h}{2}[\mu y_n +q_n]+q(t_n+\frac{h}{2})$$?

Thanks.

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#### HallsofIvy

MHF Helper
You seem to have switched from "g" to "q".

1 person

#### JhonD

Yes, you are right, it should read $$\displaystyle f(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n,y_n))=\mu y_n+\mu\frac{h}{2}[\mu y_n +g_n]+g(t_n+\frac{h}{2})$$

That would be ok?

Great, just made a numerical testing and seems to work! thanks.

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