- #1

- 559

- 8

## 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]