ODELC3

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For the first-order system $x_1'=x_2, x_2'=-\sin x_1 - x_2 |x_2| + \cos t\,$ , show that the right-hand side satisfies a Lipschitz condition in the domain $|x|\le \beta\,$ for $|t|\le \alpha\,$ , where $\alpha\,$ and $\beta\,$ are arbitrary but finite numbers. Deduce that the IVP $x_1(0)=0\,$ and $x_2(0)=0\,$ has a unique solution for $|t|\le\delta\,$ and obtain an estimate for $\delta\,$ . By allowing $\alpha\,$ and $\beta\,$ to be as large as possible, attempt to improve your estimate of $\delta\,$ .

$f(\vec x,t) = \begin{bmatrix} x_2 \\ -\sin x_1 - x_2|x_2| + \cos t \end{bmatrix}\,$

The Lipschitz constant is $L=\max\left| \frac{\partial f_1}{\partial x_1} \right| \,$ .

$L=\max\left\{ 1+|\cos x_1| + 2|x_2| \right\}\,$

$L \le 2 + 2|x_2|\,$

If $|x|\le \beta\,$ then

$L \le 2(1+\beta)\,$

The magnitude of $f\,$ gives an upper bound:

$|f(\vec x,t) = |x_2| + |\sin x_1| + |x_2^2| + |\cos t|=\beta^2+\beta+2=M\,$

Let $\delta = \min\left\{\alpha, \frac{\beta}{\beta^2+\beta+2}\right\}\,$

Pick $\alpha=\beta=1\,$ so that $\delta=\min\left\{1,\frac{1}{4}\right\}=\frac{1}{4}\,$ .

Improve the estimate for $\delta\,$ by allowing $\alpha\,$ and $\beta\,$ to be as large as possible.

Maximize $\frac{\beta}{\beta^2+\beta+2}\,$ .

The derivative is $\frac{\beta^2+\beta+2 - \beta(2\beta+1)}{(\beta^2+\beta+2)^2} = 0\,$ .

Just set the numerator equal to 0.

$\beta^2+\beta+2 - \beta(2\beta+1)=0\,$

$-\beta^2+2=0\,$

$\beta = \sqrt{2}\,$

Now let $\alpha > \frac{\sqrt{2}}{4+\sqrt{2}}\,$ so that $\delta=\frac{\sqrt{2}}{4+\sqrt{2}}=0.2612\,$

Main Page : Ordinary Differential Equations : Lipshitz Conditions

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