I am trying to learn to apply the uniqueness and existence theorem to concrete problems.
I have attached the statements of theorems at the end of post, in case to avoid any confusion.
Let $$y'=\sqrt{x^2+y^2}, y(0)=0$$ be the given differential equation.
I have the following question?
Does uniqueness and existence theorem tell(imply) that this ODE has solution or not?
To answer this question, let's look at the existence theorem. Note that $f(x,y)$ is nothing but norm of $(x,y)$ and it is standard result that norms are continuous.
So $f(x,y)$ is continuous. Since it is continuous over whole of $\mathbb{R}$, we can take any region around $(0,0)$ say $|x|\leq a, |y|\leq b$ and there will be atleast one solution due to the existence theorem.
Now I want to describe all points where IVP has a solution.
Since we know that function $f(x,y)$ is continuous on whole of $\mathbb{R}$, solution will exist for all values of $x$.
Last question is I want to use uniqueness theorem to comment on " what are all points where IVP has a unique solution"
How to solve it. I am not able to proceed?
Also please look at my work on question $1$ and $2$?