I have just been introduced to the Uniqueness and Existence theorems for first order differential equations. I can apply their methods and solve the required exercises but I am still left with a lot of basic questions.
For the purposes of this post I will focus only on Picard-Lindelöf' Theorem and it will be referred as (A).
Here is what I believe I have understood:
• (A) is used to determine if there exists a solution for a differential equation of a specific form ($dy/dx = f(x,y)$)
• (A) is used to determine whether solutions of that differential equation in a specific domain are unique (ie they do not intersect)
Here is what I am not sure I have understood:
• In order to apply the method of (A) for finding the general solution for the ODE, in a certain domain, an analytical solution has to first be accomplished so as to find a function that passes through the set of initial coordinates.
• The result of this method is the result of the analytical solution
Things I am certain I have not understood:
• When using the method of (A) it requires us to find a function that passes through the initial coordinates. Can this function be any function we want or do there have to be any other conditions than it being continuous? I understand it would help to be some function for which the convergence of said series would be easier to compute. I am asking strictly for limitations on choices, that otherwise would break the theorem.
• Why if two solutions intersect does it imply that $\partial f / \partial y$ is not continuous? My question is about intuition, I am trying to understand it graphically in the 3-axis plot of $x , y , dy/dx$
Any and all help would be very much appreciated.
EDIT 1: As per the comments I will now make clear what I mean by "method of (A)"
In the book I am studying from, after the theorem a mathod for determining the general solution, aparently based on said was introduced, namely:
$$y_n(x) = η + \int_ξ^xf(t,y_n-1(t))dt \qquad,where \; y_0(ξ) = η$$
Part of my question was about finding $y_0(x)$. I the book the differential equation is first solved analytically and then the general solution obtained, is used as $y_0(x)$.