Two (probably stupidly simple) differential equations questions

Hello!

I never thought that there would be a math forum that would be this busy! Bookmarked.

Ok... enough with the sugar. (Wink) I have finished almost all of my homework sheet, but I am having problems with a couple questions. My text (First course in DE, by Zill) is not helping me at all, and neither is the almighty google.

The first is asking me to find a differential equation that is satisfied by the function

I'm guessing it has to do with superposition, but I only know how to use that to get a solution from a DE, not vice versa!

Next up, I have a question asking me to find the intervals where these DEs have a unique solution:

a)

and

b)

EVERY question I've seen for finding intervals with a unique solution atleast has initial values.

I have been (Headbang), and of course, the math help centre at my school only has a DE assistant when I am at work, so any help you guys give me would be frickin awesome.

Thanks!