Integral approximation with midpoint method

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Dana Chica
Dana Chica 2017-3-13
I want to write a code for the Integral approximation with the midpoint method.
Mathematically, I was thinking like this: y'(t)=f(t,y(t))=-y(t)
The rectangle rule: y(t + h)=y(t) + h · f(t+h/2,y(t)+h/2*f(t,y(t)))
for:
h=0.5 and y(0)=1 (t0=0, y(t0)=1)
I would like to calculate the next step: t1=t0+h=0.5, y(t1)=?
y(t1)=y(t0+h)=y(t0)+h*f(t0+h/2,y(t0)+h/2*f(t0,y(t0)))=
=1+0.5*f(0.25,1+0.25*f(0,1))=
=1+0.5*f(0.25,1+0.25*(-1))=
=1+0.5*f(0.25,0.75) = 1+0.5*(-0.75)=0.625
I don't know how to represent the function f in Matlab (syms ?) so that it would know to calculate f(0,1) for example.
Can someone help me, please?

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