I am supposed to find a series solution for y''=y^2.
My problem is with the y^2. I've started with the following but then I get stuck:
y= sum(anx^n) and y''=sum(n*n-1)anx^n-2 and then I put these into the equation.
With the y^2 I end up with the multiplication of two series...I don't know if this is correct, and I don't know how to deal with the product of two series....
wow...that was fast. I've been working on this problem for 2 days. I have more questions if you are willing!
I am supposed to show that the solution of x' = 1 + z^4 + x^2
with x(0)=0 cannot be bounded for all z. I'm guessing I'm supposed to show that it doesn't satisfy a Lipschitz condition but I have no idea how to do that.
Thanks again...where have you been all my life.
Anyway, let f(z,x) such that :
Now you have to prove that f is Lipschitz wrt to x, uniformly to z.
So let's consider x and y. Does there exist a constant C such that ?
From the mean value theorem, we know that there exists such that :
So if you can't bound that partial derivative, then it's not a Lipschitz function.
And over , this is not bounded.