In my differential equations class, I must do the following: Thanks! Prove that

Question

In my differential equations class, I must do the following:

Thanks!

Prove that r_j(t) greaterthanorequalto -MK^j(t - t_0)^j+1/(j + 1)!, j = 0, 1, 2, ...; t [t_0 - alpha, t_0]. (Recall that this is needed in the proof of the existence theorem discussed in class.) For the following problem, recall that the local existence and uniqueness theorem for a scalar equation y' = f(t, y) assumed that f and partial differential f/partial differential y are continuous on a closed rectangle R. If instead f and partial differential f/partial differential y are continuous on a more general region, then for any point (t_0, y_0) D, we can always find a rectangle R lying completely within D. So, we can extend the theorem to the region D.

Explanation / Answer

We can extend the theorem over whole domaid D if f is linear in y,

o.w. nothing definite could be said

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