1. ## Help with antidifferentiation

Hello, I need help with the antidifferentiation of this question:

$\displaystyle \frac{dW}{dt} = (m-n-kW) \cdot W$

$\displaystyle \frac{dt}{dW} = \frac{1}{(m-n-kW) \cdot W}$

What I need to know is how to antidifferentiate(or integrate) with respect to W.
After that, I can solve the rest.
Although if you must know,
m=0.10, n=0.06, k = 0.00005
Many thanks!!!

2. Hi mrnacho

What are m, n, and k?

Btw, you can learn how to type math equation here : http://www.mathhelpforum.com/math-he...-tutorial.html

3. m, n and k are to be ignored.
What I need to know is how to antidifferentiate(or integrate) with respect to W.
After that, I can solve the rest.
Although if you must know,
m=0.10, n=0.06, k = 0.00005

4. Hi mrnacho

Ok, so m, n, and k are constants. It's really important information

So :

$\displaystyle \frac{dW}{dt}=(m-n-kW)*W$

$\displaystyle \frac{dt}{dW}=\frac{1}{(m-n-kW)*W}$

$\displaystyle dt=\frac{1}{(m-n-kW)*W}dW$

You can use partial fraction to solve this :

$\displaystyle dt=\left(\frac{A}{m-n-kW}+\frac{B}{W}\right)dW$

5. Yeah, that makes sense. But I kinda forgot how to solve this. I did separate them and then, I got lost.
1= AW + B(m-n-kW)
And I havent even anti differentiate it yet :/

6. Hi mrnacho

Of course you haven't done the integration, because you have to find the value of A and B first.

Hope this help : Partial-Fraction Decomposition: General Techniques

7. This is a big more complex but finally, my value of A is
$\displaystyle \frac{k}{m-n}$

and my value of B is:

$\displaystyle \frac{1}{m-n}$

8. Now Im stumped after getting the values and subbing, this is what I get...

$\displaystyle \frac{dt}{dW}=\frac{k}{m^2+n^2+nkW+mkW-2mn}+\frac{1}{Wm-Wn}$

How do I integrate that with respect to W?
The numbers look so scary....

9. Hi mrnacho

$\displaystyle dt=\left(\frac{A}{m-n-kW}+\frac{B}{W}\right)dW$

$\displaystyle dt=\left(\frac{\frac{k}{m-n}}{m-n-kW}+\frac{\frac{1}{m-n}}{W}\right)\; dW$

$\displaystyle \int dt=\int \left(\frac{\frac{k}{m-n}}{m-n-kW}+\frac{\frac{1}{m-n}}{W}\right)\; dW$

Because A and B are constants, it can be taken out from the integral.

$\displaystyle t=\frac{k}{m-n} \int \frac{dW}{m-n-kW}+\; \frac{1}{m-n}\int \frac{dW}{W}$