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Differential Equations

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Please see the attached file for the fully formatted problems.

(i) Consider the differential equation:
x. = x^2 , x(0) given x(0)>0

Find the solution of x(t) of this equation in terms of x(0) and show that there is a T, which depends on x(0), such that lim x(t) = infinity t --> T-

(ii) Find the solution of the differential equation
x.. - x = e^-t/2 , x(0) = 0, (0) = 1,
Using the reduction of order method.

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Solution Summary

Two differential equations are solved. The differential equations for reduction of order methods are determined. Limits that go to infinity are given.

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Solution. (1) From x'=x^2, we have
Then 1/x^2dx=dt.
Integrate both sides to get
where C is a constant. According to the initial condition we know that, when t=0, x=x(0)>0. So C=-1/x(0). The ...

Solution provided by:
  • 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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