First order differential problem question

[cupcake]

question :

As the salt KNO3 dissolves in methanol, the number x(t) of grams of the salt in a solution after t seconds satisfies the differential equation dx/dt = 0.8x - 0.004x^2

if x=50 when t=0, how long will it take for an additional 50g of salt to dissolve.

ok, here I'm encountering a problem in the differential part, I don't know how to solve the differential question.. so far, what I have done...

using separable differential equations..

1/(0.8x - 0.004 x^2) dx = dt

and integral both sides...

1/(0.8x - 0.004 x^2) dx = t+c

I don't know to integrate the right side, I use the integral calculator and the answer is

{ 5 ln x - 5 ln (x-200) } / 4 so, the whole equation will be

5 ln x - 5 ln (x-200) = 4 (t+c)

but, I think it doesn't work when I substitute x=50 when t=0. cause the ln (x-200) can't be minus..

so, please advise me

thanks

 

[Fightfish]

When you perform an integral to get a ln function, it is always ln of the absolute value of the variables. So your equation should read 5 ln |x| - 5 ln |x-200} = 4 (t+c).