# Use the Uniqueness Theorem for the Initial Value Problem

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4. For the initial value problem dy/dx = 3y^(2/3), y(2) = 0,

(a) does existence uniqueness Theorem 1 imply the existence of a unique solution? Explain.

(b) Which of the following functions are solutions to the above differential equation? Explain.

(b_1) y(x) = 0

(b_2) y(x) = (x - 2)^3

(b_3) y(x) = (x - alpha)^3, x < alpha

0, alpha <= x <= Beta

(x - Beta)^3, x > Beta

where alpha < 2 < Beta.

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This solution shows how to use the uniqueness theorem to find the solution to an initial value problem in an attached Word document.

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Solution. (a) We write this differential equation as follows.

Then we know that and are continuous at any region S of the following form where

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###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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