Check Differential Equation Solution

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Homework Help Overview

The discussion revolves around verifying a solution to a differential equation given by dX/dt=(2-x)(1-x). The original poster presents a proposed solution involving a logarithmic expression and seeks assistance in confirming its validity.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to differentiate the proposed solution and substitute it back into the original differential equation but encounters difficulties in the setup. Other participants suggest rewriting the derivative and finding a common denominator to simplify the expression.

Discussion Status

Participants are actively engaging in the verification process, with some providing guidance on algebraic manipulation. The original poster acknowledges a mistake in their calculations and appears to arrive at a conclusion about the validity of the solution, though the discussion reflects ongoing exploration of the problem.

Contextual Notes

There is mention of a common denominator that initially caused confusion, as well as the need to correctly set up the equation for verification. The discussion highlights the importance of careful algebraic handling in the context of differential equations.

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Homework Statement


Verify Differential Equation Solution
dX/dt=(2-x)(1-x)

Homework Equations


The solution is ln ((2-x)/(1-x))=t


The Attempt at a Solution


The derivative of the solution is -1/(2-x) dX/dt - (-1)/(1-x) dX/dt=1
But I plug this derivative in and I get stuck. How am I supposed to plug the solution and the derivative of the solution into the original differential equation? Can you help me set this up? I'm going to need a little help along the way...
 
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-1/(2-x) dX/dt - (-1)/(1-x) dX/dt=1

rewrite it as

\left( \frac{-1}{2-x}+ \frac{1}{1-x} \right) \frac{dX}{dt}=1

and bring the two terms in the bracket to the same common denominator.
 
Thanks for the reply. I did that and my common denominator is (2-x).

With that, I solve for dX/dt and I get dX/dt=1/(2-x).

I plug the dX/dt into the original equation which looks like 1/(2-x)=(2-x)(1-x)

the (2-x) cancels from both sides and I am left with 1=1-x and then x=0.

I went wrong somewhere...
 
I apologize...I found my mistake. The common denominator is
Code:
X[SUP]2[/SUP]-3x+2
which of course factors to (2-x)(1-x). I then multiply the numerator and denominator by the common denominator and put everything together. I then multiply both sides by (2-x)(1-x) to solve for dX/dt which leaves me with dX/dt=(2-x)(1-x). Plugging this into the original DE leaves me with (2-x)(1-x)=(2-x)(1-x) which proves that the solution is a true solution for this DE. Thanks for all of your help...
 

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