Two questions; the first is more of a check with some minor help:

1.) From calculus, what function(s) do you know where its 2nd deriv. is itself? What function(s) do you know where its 2nd deriv. is the negative of itself? Write each of the answers from above in the form of a diff. eq. with a solution.

Well, I know if f(x) = e^(x), then f''(x) = e^(x). And, I know if f(x) = cos(x), then f''(x) = -cos(x).

I'm not really sure of any other functions for those, unless you use f(x) = sin(x) for the second part, then f''(x) = -sin(x).

I guess you could use f(x) = e^(-x); then f''(x) = e^(-x)

I'm not sure how to answer these in the form of a diff. eq. with a solution unless it looks something like:

d^2y/dx^2 = -x where x = e^(x)... x = cos(x)... etc...? Uh, something like that? I'm not sure...

2.) I'm not sure where to begin on this one:

Find the region of the ty-plane where the diff. equation has a unique solution whose graph passes through a point (t_0, y_0) in that region.

(1 + y^3)y' = t^2