Answer :

I find the wording a bit odd as well, so I'll find the explicit solution first and then show implicitly that this solution does satisfy the given DE. 

Starting with ln[(2X - 1)/(X - 1)] = t, solve for X in terms of t: 

(2X - 1)/(X - 1) = e^t ----> 

2X - 1 = (X - 1)*e^t ----> 

2X - X*e^t = 1 - e^t -----> 

X*(2 - e^t) = 1 - e^t -----> 

X = (1 - e^t)/(2 - e^t) = (e^t - 1)/(e^t - 2). 

Now differentiate ln[(2X - 1)/(X - 1)] = ln(2X - 1) - ln(X - 1) = t implicitly: 

(2/(2X - 1))*dX/dt - (1/(X - 1))*dX/dt = 1 ------> 

dX/dt*((2*(X - 1) - (2X - 1)) / ((2X - 1)(X - 1))) = 1 ----> 

dX/dt*(-1) = (2X - 1)(X - 1) -----> 

dX/dt = (X - 1)(1 - 2X).

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lhmarianateixeira

In this exercise we have to use the knowledge of implicit derivative to calculate the value of the given equation:

[tex]dX/dt = (X - 1)(1 - 2X)[/tex]

To start the exercise we have to remember how to do an implicit derivation, so:

The implicit derivation is a method that allows us to find the derivative of a function that is only shown implicitly to us, like this y=f(x). In this case, we don't have exactly the function, but a relation that mixes x with y.

First, we have that the equation is given by:

[tex]ln[(2X - 1)/(X - 1)] = t\\(2X - 1)/(X - 1) = e^t\\2X - 1 = (X - 1)*e^t\\2X - X*e^t = 1 - e^t \\X*(2 - e^t) = 1 - e^t \\X = (1 - e^t)/(2 - e^t) = (e^t - 1)/(e^t - 2)[/tex]

Now using the same equation and differentiating it, we find that:

[tex]ln[(2X - 1)/(X - 1)] = ln(2X - 1) - ln(X - 1) = t \\(2/(2X - 1))*dX/dt - (1/(X - 1))*dX/dt = 1\\dX/dt*((2*(X - 1) - (2X - 1)) / ((2X - 1)(X - 1))) = 1 \\dX/dt*(-1) = (2X - 1)(X - 1) \\dX/dt = (X - 1)(1 - 2X)[/tex]

See more about implicit derivate at brainly.com/question/24516698

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