nonlinear_example

nonlinear_example - An Example of a Non-linear Differential...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: An Example of a Non-linear Differential Equation R. C. Daileda In class we mentioned the following theorem, whose proof the interested reader can find in Section 2.8 of [1]. Theorem. 1. Consider the initial value problem (IVP) y ′ = f ( t, y ) , y ( t ) = y . (1) If f and ∂f/∂y are both continuous on a disk centered at ( t , y ) then (1) has a unique solution defined on some interval t − h < t < t + h , h > . Several points should be made here. While this theorem does give us an effective way of determining if a solution to an IVP exists, it gives us no way of determining this solution nor does it give any information on the interval of definition of that solution. This should be compared to the analogous fact for IVPs involving linear first order ODEs (Theorem 2.4.1 of [1]). If f is linear then we can write down an explicit solution, and the interval of definition of that solution can be determined from f and t alone . The moral is that the behavior of solutions to non-linear differential equations can be drastically different than that of linear equations, as the following example is meant to illustrate....
View Full Document

Page1 / 3

nonlinear_example - An Example of a Non-linear Differential...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online