Note: When the textbook says ‘f satifies a Lipschitz condition’, it implies that f is Lipschitz continuous in the dependent variable y, not the independent variable x.
1. Section 70, Problem 3
2. Section 70, Problem 5
3. Determine if this function is Lipschitz continuous (specify the domain)
f (x) = −x1 + x1x2
x2 − x1x2
4. Prove that, if f, h : R ! R are locally Lipschitz over some bounded domain D, then f + h, f h, and f ◦ h are locally Lipschitz.
Application of Picard’s theorem:
5. Section 70, Problem 6. (Note: the DE is y0 = |y|.)
6. Section 70, Problem 7.
7. Does Picard’s theorem apply to the following IVP, if yes, what does it imply? What is maximal interval of existence of the solution; you may find this by evaluating the explicit solution. Observe how the interval of existence changes with y0.
dydt = ty3, y(0) = y0
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