Consider the following one-dimensional system, dx/dt = F(x), where F : (0,∞) → R is given by, $$F(x) = 1\div2x$$ ∀x > 0. For p > 0, find the solution, u : J → R, of the above ODE, subject to the following initial condition: x(0) = p
Find the maximal interval of existence, J, for the solution, u, computed in part (a) of this problem. Compute, $$\ \lim_{t\to -(p^2)^+}u(t)$$
Let F be as in problem 1. Denote by θ(t, p) the solution u(t), ∀t ∈ J of the IVP given in problem 1. Find the domain of the definition of θ and verify that θ is continuous on its domain.
I solved the first problem and the result was $$u(t)=({\sqrt {(t+p^2})}$$ Now I am confused between the maximum interval of existence and the domain of definition. I know J is an interval and it must contain the initial time and domain contains both (t,p). $$θ:I×U → U$$ U is an open subset of R and D⊆I×U and J⊆I containing t. Here's what I did $$J=(-p^2,\infty)$$ and $$\ \lim_{t\to -(p^2)^+}u(t)=0$$. And $$D=(0,\infty)$$ So am I correct? Also how should I show θ is continuous? Do I have to show that F(x) satisfies the Lipshcitz condition or can I use the limits to show it is continous. I also have to show θ is a C^1 flow map in its domain.