# ODELC3

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 $|x|\leq \beta \,$ for $|t|\leq \alpha \,$, 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 $|t|\leq \delta \,$ 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 $L=\max \left|{\frac {\partial f_{1}}{\partial x_{1}}}\right|\,$.

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

$L\leq 2+2|x_{2}|\,$

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

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

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 $\delta ={\frac {{\sqrt {2}}}{4+{\sqrt {2}}}}=0.2612\,$