hw7 - Math 425, Spring 2011 Jerry L. Kazdan Problem Set 7...

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: Math 425, Spring 2011 Jerry L. Kazdan Problem Set 7 DUE: Thursday March 24 [ Late papers will be accepted until 1:00 PM Friday ]. 1. Suppose u is a twice differentiable function on R which satisfies the ordinary differential equation u 00 + b ( x ) u- c ( x ) u = , where b ( x ) and c ( x ) are continuous functions on R with c ( x ) > 0 for every x ∈ ( , 1 ) . a) Show that u cannot have a positive local maximum in the interval ( , 1 ) , that is, have a local maximum at a point p where u ( p ) > 0. Also show that u cannot have a negative local minimum in ( , 1 ) . [The example u 00 + u = 0 has u ( x ) = sin x as a solution, which does have posivive local maxima and negative local minima. This shows that some assumption, such as our c ( x ) > is needed.] b) If u ( ) = u ( 1 ) = 0, prove that u ( x ) = 0 for every x ∈ [ , 1 ] . c) If u satisfies 4 u xx + 3 u yy- 5 u = in a region D ⊂ R 2 , show that it cannot have a local positive maximum. Also show that u cannot have a local negative minimum.cannot have a local negative minimum....
View Full Document

This document was uploaded on 03/06/2012.

Page1 / 2

hw7 - Math 425, Spring 2011 Jerry L. Kazdan Problem Set 7...

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