How to solve this diffrential equation using parts formula?
Proving single solution to an initial value problem
or each I need to prove that there is a single solution defined on
Continuing Solutions of to entire number line
can be continued to the whole real line.
I know that this ODE is seperable as follows
Thus, giving the solution
However, forom here, it is not clear to me how any solution x(t) can be continued to the entire real number line.
How do I get an estimate for this nonlocal ODE?
Consider the following nonlocal ODE on :
where l is a positive integer and is a real number.
Define the following norm
I want to prove the estimate:
for some constant C independent of , l and f. But I am stuck.
Here is what I tried. Multiply both sides by f and integrate by parts to get:
where I used Cauchy-Schwartz in the before last line. I am not sure how to continue and how to get rid of the f'(1) term.
Any help is appreciated.