Verify implicit solution
dx/dt=(x-1)(2x-1), ln(2x-1/x-1)=t

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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