Finish this problem? (Diff Eq)

[iRaid]

Homework Statement

$$\frac{dx}{dt}=x-x^{2}$$

Homework Equations

The Attempt at a Solution

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

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

 

[ehild]

$$x=De^{t}(x-1)$$

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

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

 

[verty]

It was 95% complete, just to collect terms.

 

[iRaid]

I don't know why I didn't see that before... Maybe I was just too tired.. Thank you.