Discussion

For the differential equation :

\[ y' = x \sin y + y \cos x \]

defined on rectangle

\[ |x| \le 5, \; |y| \le 2, \]

which of the following is Lipschitz constant ?

Answer: B

A Lipschitz constant is 6.

For \(f(x, y) = x \sin y + y \cos x\), the Lipschitz constant in y on the rectangle is the bound of \(|\partial f/\partial y|\):

\[ \begin{aligned} \left|\frac{\partial f}{\partial y}\right| &= |x \cos y + \cos x| \\ &\le |x| + 1 \le 5 + 1 = 6 \end{aligned} \]

By the mean value theorem, \(|f(x, y_{1}) - f(x, y_{2})| \le 6|y_{1} - y_{2}|\) on \(|x| \le 5\), \(|y| \le 2\).

The smaller values do not work: at \(x = 5\), \(y = 0\), the derivative is \(5 + \cos 5 \approx 5.28\), already larger than 4, 3 or 1.

No comment is present. Be the first to comment.

Loading…