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http://www.math.uiuc.edu/~tyson/existence.pdf (example 5)

In the example, the solution for the interval is $\delta_2 = |x_0|$. I'm confused on how they got that. What was the algebraic step to solve for $\delta_2$? I mean I can reason it out, and it makes sense. However, how do I actually solve for it algebraically? Thank you!

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2 Answers 2

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We know that an ODE of the form $$y' = F(x,y),\quad y(x_{0}) = y_{0}$$ is guaranteed a unique solution when both $F(x,y)$ and $F_{y}(x,y)$ are continuous. For Example $5$, $$y' = 2y/x,\quad y(x_{0}) = y_{0},$$ we see that both $$F(x,y) = 2y/x\quad \text{and}\quad F_{y}(x,y) = 2/x$$ are continuous so long as $x\neq 0$. That means that first of all we get an interval of uniqueness for some $\delta_{2} > 0$: $$ x_{0} - \delta_{2} < x < x_{0} + \delta_{2}.$$ Because we need to keep $x$ away from $0$, and we don't know if $x_{0}$ is greater than or less than $0$ we have two cases:

First, if $x_{0} > 0$ we can take $$x_{0} - \delta_{2} = 0 \implies \delta_{2} = x_{0}.$$

Second, if $x_{0} < 0$ we can take $$x_{0} + \delta_{2} = 0 \implies \delta_{2} = |x_{0}|.$$

In either case, it suffices to take $\delta_{2} = |x_{0}|$ to guarantee uniqueness.

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  • $\begingroup$ a more general question is (from en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem) $a < min\{b/M, 1/L\}$. What is b, M, and L in this context? Thank you! $\endgroup$
    – Phu Nguyen
    Commented May 9, 2017 at 15:16
  • $\begingroup$ @PhuNguyen I clarified my answer a bit. $\endgroup$
    – DMcMor
    Commented May 9, 2017 at 15:22
  • $\begingroup$ Thank you! I understand that now, but can you please answer my more general question on the topic? Thanks! $\endgroup$
    – Phu Nguyen
    Commented May 9, 2017 at 15:25
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To satisfy conditions for uniqueness, solutions in this example have to stay away from $0$. Since the distance from a point $x_0\ne0$ to $0$ is $|x_0|$, an interval on which a unique solution exists cannot go by more than $\delta_2=|x_0|$ away from $x_0$.

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  • $\begingroup$ That's what I figured logically. This is a simple equation with a simple solution. What happens if the equation is much more complicated? I won't be able to reason out like on this problem. How do I solve for $\delta_0$ algebraically? Thank you! $\endgroup$
    – Phu Nguyen
    Commented May 9, 2017 at 15:11
  • $\begingroup$ @PhuNguyen: I'd say it's still based on the same idea -- you need to find discontinuities of $F$ and $\partial F/\partial y$ and stay away from them. So the issue is not as much an ODE problem as it is an algebra problem. I don't think we can say that there's a common method for all of them -- depends on the functions. $\endgroup$
    – zipirovich
    Commented May 9, 2017 at 15:32
  • $\begingroup$ Lol, so basically, I have to reason out logically for each individual case? If that is the case, I guess I got the idea now. Thanks! $\endgroup$
    – Phu Nguyen
    Commented May 9, 2017 at 15:57

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