Is p(r) = (2/R²)r the Only Solution to the Integral Uniqueness Problem?

  • Thread starter Thread starter HyperbolicMan
  • Start date Start date
  • Tags Tags
    Integral Uniqueness
Click For Summary

Homework Help Overview

The discussion revolves around the uniqueness of a continuous function p(r) defined on the interval [0, R], which satisfies a specific integral condition. The original poster is attempting to prove that p(r) = (2/R²)r and is questioning whether this solution is unique.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss proving the uniqueness of p(r) by exploring the implications of its deviation from the proposed solution. There is a suggestion to derive a contradiction based on the behavior of p(r) over an interval.

Discussion Status

The conversation includes attempts to clarify the uniqueness of the function p(r) and explores mathematical reasoning related to derivatives and integral properties. Some guidance has been offered regarding how to approach the proof, but no consensus has been reached on the uniqueness of the solution.

Contextual Notes

There is an emphasis on the continuity of p and the specific integral condition that must be satisfied. The original poster expresses uncertainty about proving the uniqueness, indicating a potential gap in information or understanding.

HyperbolicMan
Messages
14
Reaction score
0
Let p be a continuous function such that for all r1, r2 in [0,R], ∫r2r1p(r)dr=(r22-r12)/R2.

I'm trying to prove that p(r)=(2/R2)r.

Question: Must p be unique? I'm not sure how to prove/disprove this.
 
Physics news on Phys.org
Try showing that p(r) - 2r/R^2 must be zero. Hint: Prove that if it isn't zero, there is a nondegenerate interval [r_1, r_2] on which it is either strictly positive or strictly negative, and derive a contradiction.
 
HyperbolicMan said:
Let p be a continuous function such that for all r1, r2 in [0,R], ∫r2r1p(r)dr=(r22-r12)/R2.

I'm trying to prove that p(r)=(2/R2)r.

Question: Must p be unique? I'm not sure how to prove/disprove this.

Let r1 be constant and r2 variable. Take the derivative of the integral [(r22-r12)/R2]
with respect to r2 and rename r2 to be r.

The derivative is unique.
 
Thanks for the help!
 

Similar threads

Calculate this Integral around the Circular Path using Green's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
Show uniqueness in Dirichlet problem in unit disk
  • · Replies 6 ·
Replies
6
Views
2K
Proving Irregularity of {(x, y, z) within R^3 : x^2 + y^2 - z^2 = 0}
  • · Replies 14 ·
Replies
14
Views
4K
Symmetry of an Integral of a Dot product
  • · Replies 9 ·
Replies
9
Views
2K
Existence and Uniqueness For ODE
  • · Replies 6 ·
Replies
6
Views
2K
Setting the limits of an integral
  • · Replies 3 ·
Replies
3
Views
2K
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
  • · Replies 7 ·
Replies
7
Views
2K
Dynamical Systems: how to find equation for Poincare map?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
3K
Is the set open, closed, neither, bounded, connected?
  • · Replies 7 ·
Replies
7
Views
3K