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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|\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\,

Main Page : Ordinary Differential Equations : Lipshitz Conditions