Picard existence theorem and IVP

[complexnumber]

Homework Statement

Consider the initial value problem
<br /> \begin{align*}<br /> \left\{<br /> \begin{array}{l}<br /> \displaystyle \frac{dy}{dx} = \exp(xy) \\<br /> y(0) = 1<br /> \end{array}<br /> \right.<br /> \end{align*}<br />

1. Verify that this IVP has a unique solution in a neighborhood
of x = 0.

2. Following the notation of the lectures, find the values of K,
M, and \delta that will work for this case.

Homework Equations

The Attempt at a Solution

1. Let \displaystyle f(x,y) = \frac{dy}{dx} = \exp(xy). Then
\displaystyle \frac{df}{dy} = x \exp(xy) which is continuous and
hence has upper bound K. Hence according to Picard's theorem the
IVP has a unique solution in \abs{x - x_0} \leq \delta.

2. How can I find the values of M, K and \delta? Does M mean the upper bound of function f(x,y)?

 

[ystael]

2. Following the notation of the lectures, find the values of K,
M, and \delta that will work for this case.

2. How can I find the values of M, K and \delta? Does M mean the upper bound of function f(x,y)?

We cannot possibly help you with this unless you explain "the notation of the lectures".

 

[complexnumber]

We cannot possibly help you with this unless you explain "the notation of the lectures".

I think \delta relates to the neighborhood |x - x_0| &lt; \delta where the differential equation has a unique solution. K is the constant in Lipschitz condition. M is the upper bound of function f(x,y).