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Solving Differential Equations with Substitution and Bernoulli

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56. Suppose that n does not equal to zero and n does not equal to one. Show that the substitution v = y1-n transforms the Bernoulli equation dy/dx + P(x)y = Q(x)yn into the linear equation dv/dx + (1-n)P(x)v(x) = (1-n)Q(x).

63. The equation dy/dx = A(x)y2 + B(x)y + C(x) is called a Riccati equation. Suppose that one particular solution y1(x) of this equation is known.
Show that the substitution y = y1 + (1/v) transforms the Riccati equation into the linear equation dv/dx + (B +2Ay1)v = -A

66. An equation of the form y = xy' + g(y') is called a Clairaut equation. Show that the one parameter family of straight lines described by y(x) = Cx + g(C) is a general solution of y = xy' + g(y').

keywords: ODE, ODEs

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56. Suppose that n does not equal to zero and n does not equal to one. Show that the substitution v = y1-n transforms the Bernoulli equation dy/dx + P(x)y = Q(x)yn into the linear equation dv/dx + (1-n)P(x)v(x) = (1-n)Q(x).

Proof. Let v = y1- n. Then by the chain law, we can have

So,
..........................(1)
Note that
......................................(2)

We multiply by y- n on both sides of the Bernoulli equation dy/dx + P(x)y = Q(x)yn, we ...

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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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