Making square you add one more root.
Hi again. Here's one.
(a) Without solving, explain why the intitial value problem
has no solution for .
(b) Solve the initial value problem in part (a) for and find the largest interval on which the solution is defined.
My attempt:
(a) Because the only way to arrive at a radical through differentiation is to begin with one, and any expression involving a radical (even root) with nothing outside of the radical must always give a positive result over the reals.
(b) I get
.
If I set , I get .
But, I won't evfen attempt to find the interval of definition because the book doesn't include the plus or minus. They have .
What's the deal here?
The key idea is that the domain of your solution does not have to be the same as the interval of existence and uniqueness of the solution to an ODE.
We know that if and
are continuous and is an interior point of the above rectangle.
Then there exists an interval such that a solution exists and is unique.
Now for your problem is discontinous at
Now given an initial point then the solution "is"
Things are already starting to look off because now we have a non unique solution and here is the kicker notice what happens when
This gives Now I can make another solution as follows
Notice this new function is continuous and differentiable on all of but is not unique. So we run into problems where is undefined.
So the largest interval with a unique solution is either