Problem:
Reconsider the fundamental theorem of calculus as expressed in (1.1.16 found in the lecture note packet). But now, integrate from $t-\Delta t$ to $t+\Delta t$. Using (1.1.19), what is the new version of (1.1.20):
Solution:
We start with $ y(t + \Delta t) - y(t - \Delta t) = \int\limits_{-t - \Delta t}^{t + \Delta t}{f(t, y)dt} $.
Next we approximate the integral by constant and put terms around, we get
$ y_{n+1} = y_{n-1} + 2\Delta t f(t, y_n) $.
Reconsider the fundamental theorem of calculus as expressed in (1.1.16 found in the lecture note packet). But now, integrate from $t-\Delta t$ to $t+\Delta t$. Using (1.1.19), what is the new version of (1.1.20):
Solution:
We start with $ y(t + \Delta t) - y(t - \Delta t) = \int\limits_{-t - \Delta t}^{t + \Delta t}{f(t, y)dt} $.
Next we approximate the integral by constant and put terms around, we get
$ y_{n+1} = y_{n-1} + 2\Delta t f(t, y_n) $.
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