(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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]?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Picard existence theorem and IVP

**Physics Forums | Science Articles, Homework Help, Discussion**