Picard existence theorem and IVP

[complexnumber]

Homework Statement

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

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

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

Homework Equations

The Attempt at a Solution

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

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

 

[ystael]

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

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

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 [tex]\delta[/tex] relates to the neighborhood [tex]|x - x_0| < \delta[/tex] where the differential equation has a unique solution. [tex]K[/tex] is the constant in Lipschitz condition. [tex]M[/tex] is the upper bound of function [tex]f(x,y)[/tex].