Homework 8 - Bonus Problem <me@tylerlogic.com>
November 8, 2014
http://coursework.tylerlogic.com/courses/upenn/math508/homework08/bonusproblem

### Problem:

B-1
Say a function u(x) satisfies the differential equation
 (1)

on the interval [0,A] and that the coefficients b(x) and c(x) are both bounded, say |b(x)|≤ M and |c(x)|≤ M (if the coefficients are continuous, this is always true for some M).

a)
Define E(x) := (u2 + u2). Show that for some constant γ (depending on M) we have E(x) γE(x). [Suggestion; use the inequality 2xy x2 + y2.]
b)
Use Problem 3(c) above to show that E(x) eγxE(0) for all x [0,A].
c)
In particular, if u(0) = 0 and u(0) = 0, show that E(x) = 0 and hence u(x) = 0 for all x [0,A]. In other words, if u′′ + b(x)u+ c(x)u = 0 on the interval [0,A] and that the functions b(x) and c(x) are both bounded, and if u(0) = 0 and u(0) - 0, then the only possibility is that u(x) 0 for all x 0.
d)
Use this to prove the uniqueness theorem : if v(x) and w(x) both satisfy equation (1) and have the same initial conditions, v(0) = w(0) and v(0) = w(0), then v(x) w(x) in the interval [0,A].

### Solution:

B-1
a)
From equation 1 we have u′′ = -b(x)u′- c(x)u so that |u′′| = |b(x)||u′| + |c(x)||u| which implies
 (2)

Define E = (u2 + u2). Then

substituting in |u′′| from equation 2 we continue with
Since E is always positive, and E = (u2 + u2) uu then there exists an integer n so that nE ≥|uu|. We can there for continue as
Hence if we define the constant gamma as γ = 2M + n(M + 1) then we get
 (3)

b)
As an immediate consequence of this homework’s problem 3c, equation 3 implies that E(x) eγxE(0) for all x [0,A].
c)
d)