Finish this problem? (Diff Eq)

  • Thread starter iRaid
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  • #1
iRaid
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Homework Statement


[tex]\frac{dx}{dt}=x-x^{2}[/tex]


Homework Equations





The Attempt at a Solution


I think the only thing I have wrong so far is how to finish it because I cant find anything wrong with my work, but I don't know how the book gets their final answer.
Separate variables...
[tex]\int \frac{dx}{x(1-x)}=\int dt[/tex]
Partial fraction decomposition..
[tex]\int (\frac{1}{x}-\frac{1}{x-1})dx=t+C\\ln|x|-ln|x-1|=t+C\\ln|\frac{x}{x-1}|=t+C[/tex]
Using properties of e...
[tex]\frac{x}{x-1}=e^{t}e^{C}[/tex]
D=e^C (professor wants us to write it like this) and multiply by x-1 on both sides..
[tex]x=De^{t}(x-1)[/tex]

That's not the solution though and I'm not sure why.
Final answer: [tex]x(t)=\frac{C}{C-e^{t}}[/tex]
 

Answers and Replies

  • #2
ehild
Homework Helper
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[tex]x=De^{t}(x-1)[/tex]

That's not the solution though and I'm not sure why.
Final answer: [tex]x(t)=\frac{C}{C-e^{t}}[/tex]

You have to isolate x, writing the solution in the form x(t)=f(t). Expand your equation, collect the terms with x at one side, factor it ....

The final solution you quoted is not correct. There must be a minus in front of t in the exponent.

ehild
 
  • #3
verty
Homework Helper
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It was 95% complete, just to collect terms.
 
  • #4
iRaid
559
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I don't know why I didn't see that before... Maybe I was just too tired.. Thank you.
 

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