Solutions to differential equations

Section 25.6 Solutions to differential equations

Define sequential continuity for functions on \(\bb{R}^2\text{.}\)
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Construct approximate solutions to differential equations using Picard iteration.
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Prove the Picard-Lindelöf theorem to establish existence and uniqueness of solutions to initial value problems.
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Introduction goes here.

Sequential definition of continuity in 2D: Let \(U\subseteq\bb R^{2}\) be a set, \(F:U\to\bb R\) be a function, and \((x,y)\in U\) be a point. The function \(F\) is continuous at \((x,y)\) if for every sequence \(((x_{n},y_{n}))_{n=1}^{\infty}\) of points in \(U\) such that \(\limit{n\to\infty}x_{n}=x\) and \(\limit{n\to\infty}y_{n}=y\text{,}\) we have

\begin{equation*} \limit{n\to\infty}F(x_{n},y_{n})=F(x,y)\text{.} \end{equation*}

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Theorem 23. Picard-Lindelöf theorem.

Let \(I,J\subseteq\bb R\) be compact intervals, let \(I^{\circ}\) and \(J^{\circ}\) be their interiors, and let \((x_{0},y_{0})\in I^{\circ}\times J^{\circ}\text{.}\) Suppose \(F:I\times J\to\bb R\) is continuous and Lipschitz in the second variable, that is, there exists an \(L\in\bb R\) such that

\begin{equation*} \abs{F(x,y)-F(x,z)}\le L\abs{y-z}\qquad\text{for all }y,z\in J, x\in I\text{.} \end{equation*}

Then there exists an \(h>0\) such that \([x_{0}-h,x_{0}+h]\subseteq I\) and a unique differentiable function \(f:[x_{0}-h,x_{0}+h]\to J\subseteq R\) such that

\begin{equation*} f'(x)=F(x,f(x))\qquad\text{and}\qquad f(x_{0})=y_{0}\text{.} \end{equation*}

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In other words, initial value problems have unique solutions under suitable conditions.
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Proof relies on the idea of Picard iteration, the FTC, and uniform convergence of a sequence of functions, as well as the uniform norm.
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