- #1

Daxin

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- 0

## Homework Statement

Suppose u, v are two linearly independent solutions to the differential equation u''+p(x)u'+q(x)v=0. If x0,x1 are consecutive zeros of u, then v has a zero on the open interval (x0,x1)

## Homework Equations

## The Attempt at a Solution

I'm trying to use the Wronskian(u,v;x) to arrive at a contradiction.

I know that the W = uv'-u'v is never 0 since u,v are linearly independent. Furthermore, since it is a combination of C2 functions it must also be continuous on [x0,x1] = I. Also if I suppose that v is never zero on I, then it must be always positive or always negative. Same with u, since we know x0,x1 are consecutive zeros. From this I want to show that the Wronskian is positive at some point in I, and negative at another point, which would lead to a contradiction. Is this the right idea?