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I need to plot this simple system:

x'' = -x

using midpoint Euler.

u1 = -x , u2 = -x'

u1' = u2

u2' = -x

u1(n+1) = u1(n) + h*?

u2(n+1) = u2(n) + h*f((1/2)*(u1(n) + u1(n+1))

We dont know u1(n+1). I tried approximating it with u1(n+1) = u1(n) + h*u2(n)

u2(1+i) = u2(i) + h * ((-1/2) * (u1(n) + u1(n+1))

Then we have u2(i+1) and u2(i). A midpoint value is (u2(1+i) - u(i))/2

u1(q+i) = u1(i) + h*midpoint

When i plot this the result is some awful diverging line, not an oscillating function. What is wrong?

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# Midpoint Euler method, second order system

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