Determine the largest interval where there is guaranteed to be a unique solution for the following IVP. Justify why this is the largest interval.
Alright, well I've put some thought into this problem. I also asked my prof about it. What I found interesting in her reply was that she said I should look at theorems and that I DONT have to solve a system to know whether a unique sol'n exists. Looking at your response, we're given the derivative. That is, X' = ... .
Looking at theorems in my book, this may be helpful:
(Note that the form I gave was X' = AX + F(t)).
So, the theorem says to let the entries of the matrices A(t) and F(t) be functions that are continuous on a common interval I that contains the point t_0. Then there exists a unique sol'n to the IVP X' = A(t)X + F(t) subject to X(t_0) = X_0 on the interval.
So, I'm not quite sure exactly on how to go about this...
I decided to look at the functions within the matrix. That is, we know ln(t) is defined for t = (0, infinity). And, 3t^2 is defined for all reals, along with sin(t). Now we look at F(t).
sqrt(5 - t) is defined for t = (-infinity,5]. And cot(t) is defined (**not really sure.. every multiple of pi/2??)..
So now we have to see which interval 5pi/2 exists (the largest one?).
Sorry a little confused. Does my reasoning sound kind of right? I'd appreciate any help.
Well, I'm starting to think no such interval exists. B/c when I tried getting a hint I was told: "Find the interval (if it exists) that has the initial condition AND all the constraints" something along those lines... tricky hard problem. Wish I could just solve this.