# 12 are unique as expressed by the following theorem

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ) be continuous and satisfy a Lipschitz condition on f(t y) j 0 t T ;1 < y < 1g: Then the IVP (1.1.2) has a unique continuously di erentiable solution on 0 T ] for all y0 2 (;1 1). Proof. The proof appears in most books on ODE theory, e.g., 2]. Remark 1. A Lipschitz conditions guarantees unique solutions and is not needed for esistence. For example, the IVP y = 3y2=3 y(0) = 0 0 has the two solution y(t) = 0 and y(t) = t3 . This f (t y) = 3y2=3 does not satisfy a Lipschitz condition. 9 Remark 2. Theorem 1.2.1 applies when f satisfys a Lipschitz condition on a compact, rather than an unbounded, domain D as long as the solution y(t) remains in D 2]. In addition to existence and uniqueness, we will want to know something about the stability of solutions of the IVP. In particular, we will usually be interested in the sensitivity of the solution to small changes in the data. Perturbations arise naturally in numerical computation due to discretization and round o errors. A formal study of sensitivity would lead us to the following notion of a well-posed...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern