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Differential equation and explicit solution

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The problem is:

dx/dt = (x-1)(1-2x); ln {2x-1/x-1} = t

To solve this problem I must, first, verify that the indicated expression is an implicit solution of the given first-order differential equation then find at least one explicit solution. The last element of the solution is to use a graphing utility to obtain the graph of an explicit solution and give an interval of definition.

I have been working on this problem for quite a while and have been unable to come up with the solution given in the book. This solution is:

X = e^t - 1 / e^t-2 defined on (-infinity, ln2) or on (ln2, infinity)

Please supply me with a detailed answer to this problem so that I will be able to use the principles to solve more of the same type problem.

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Solution Summary

This provides an example of solving a differential equation for an explicit solution and to use a graphing utility to obtain the graph.

Solution Preview

first, we verify that the given function, defined implicitly, is indeed a solution to the differential equation:

take the exponential of both sides

(2x - 1)/(x - 1) = e^t

then differentiate both sides with respect to t, using the quotient rule on the left side:

- 1/(x - 1)^2 . dx/dt = e^t

solving for dx/dt gives

dx/dt = - e^t ( x - 1)^2

now we use e^t = (2x - 1)/(x - 1) to obtain

dx/dt = - [ (2x - 1)/(x - 1) ] (x - 1)^2

equivalently,

dx/dt = [(1 - 2x)/(x - 1)] (x - 1)^2

or

dx/dt = (1 - 2x)(x - 1)

which is what we wanted to show

to solve the equation by hand, we must separate variables:

dx/ (x - 1)(1 - 2x) ...

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