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
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.
So you know that the general solution to the DE is:
Also you have the initial condition , but:
So , and the solution is: