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Verify an expression is an implicit solution to a first order DE.

  • Thread starter oddjobmj
  • Start date
  • #1
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Homework Statement


Verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution y=∅(x) in each case.


Homework Equations



[itex]\frac{dX}{dt}[/itex]=(X-1)(1-2X); ln([itex]\frac{2X-1}{X-1}[/itex])=t

The Attempt at a Solution


I know I need to solve the 't=' portion for X and here it is:

ln([itex]\frac{2X-1}{X-1}[/itex])=t =>

[itex]\frac{e^t-1}{e^t-2}[/itex]=X

In -explicit- cases I would differentiate X to find X' and replace the original X' from the given differential equation to see if it works out to 0.

I don't know how to go about proving it in the implicit case and everything I'm finding when I google is how to do implicit differentiation from a calc 2-3 perspective.

Thank you
 

Answers and Replies

  • #2
Bacle2
Science Advisor
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I don't see how you go from:

ln ( (2X-1)/(X-1))=t to (et-1)/(et -2) =X


I would think of raising e to both sides.
 
  • #3
306
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I don't see how you go from:

ln ( (2X-1)/(X-1))=t to (et-1)/(et -2) =X


I would think of raising e to both sides.
Exactly. When you do that the ln goes away and you end up with e^t on the other side. We do some algebra and end up with the above result. I have verified this with wolframalpha.

Edit: Here are the steps for the curious-

Exponentiate both sides to get: (2X-1)/(X-1)=e^t
Multiply both sides by the denominator (x-1) to get: 2x-1=xe^t-e^t
Get the components with x's in them together: xe^t-2x=e^t-1
Factor out the x: x(e^t-2)=e^t-1
Divide both sides by (e^t-2) to get x by itself: x=(e^t-1)/(e^t-2)

Edit 2:
The back of the book says that x=(e^t-1)/(e^t-2) is a solution. I have to figure out why that's the case.
 
Last edited:

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