Originally Posted by

**griffsterb** Hey guys, been looking for some online support for diff eqs and this is where I wound up :P Just started summer session last week so I have a semester's worth of diff eqs crammed into 7 weeks and the HW is coming hard and heavy. This question I have is pretty elementary stuff but I am not very well-versed in diff eqs so I am having some trouble with it.

Here it is verbatim from the book:

a.) Draw a direction field for the given differential equation. How do solutions appear to behave as t becomes large? Does the behavior depend on the choice of the initial value *a*? Let *a0* be the value of *a* for which the transition from one type of behavior to another occurs. Estimate the value of *a0*.

b.) Solve the initial value problem and find the critical value *a0* exactly.

c.) Describe the behavior of the solution corresponding to the initial value *a0*.

Equation: y' - y/2 = 2cost, y(0) = a

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Now, I can draw the direction field, albeit it's a little time-consuming. I can solve the equation as well since it's linear; not too tough. The problems I have are with the initial value stuff. I just plain don't understand how to do it.

Does the behavior depend on the initial value a? I don't know, I don't think so. I'm not sure how to tell. Estimate the value of *a0*? I have no idea how to do this either, other than looking at my direction field and just guessing. Find the critical value *a0*? Don't know how to do that one.

I mean, when you solve the equation you're going to be left with a constand C and then you have your initial value y(0) = *a *= (some equation + C). How are you supposed to solve for *a* when you have an unknown C?

Anyway, this stuff probably isn't hard but it's more or less my first experience with it, so I just need to get my mind wrapped around it a little bit I think. Hopefully someone around here can help me out. Thanks guys.