# Partial derivatives

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The heat transfer in a semi-infinite rod can be described by the following PARTIAL differential equation:

∂u/∂t = (c^2)∂^2u/∂x^2

where t is the time, x distance from the beginning of the rod and c is the material constant. Function

u(t,x) represents the temperature at the given time t and place x. Verify that the function

u(t,x) = (e^-t)(cos x/c)

is the solution of the heat equation (i.e. it satisfies the heat equation.)

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##### Solution Summary

This shows how to verify that a given function satisfies the heat equation for a specific situation.

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Proof. Denote the partial derivative of u(t,x) with respect to t by u_t(t,x), and denote the double partial derivative of u(t,x) w.r.t x by ...

###### 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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- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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