# How to solve this 2nd differential eq?

1. Aug 31, 2004

### AronH

I was making a problem about population grow, and I wasn't able to solve this:
x'=x*(M-x), for x(t).
Can anyone help me?
Thanks.

2. Aug 31, 2004

### ahrkron

Staff Emeritus
I have the impresion that there is no analytical solution to that one. IIRC, its solution has a chaotic behavior depending on M (though there may be another parameter).

3. Aug 31, 2004

### Hurkyl

Staff Emeritus
I think it's seperable.

4. Aug 31, 2004

### Tide

Integrate directly to find
$$\frac{x}{x_0} \times \frac{M-x}{M-x_0} = e^{M t}$$
from which you can find x(t) by solving the quadratic equation.

5. Sep 1, 2004

### Galileo

scribbly scribbly...

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

... doesn't work, never mind...

edit: wait,wait,wait... partial fractions:

$$\frac{1}{xM-x^2}=\frac{1}{Mx}+\frac{1}{M(M-x)}$$

$$\int \frac{dx}{Mx}+\int \frac{dx}{M(M-x)}=\frac{1}{M}(\ln(\frac{x}{x_0})-\ln(\frac{M-x}{M-x_0}))=\frac{1}{M}\ln\left(\frac{x(M-x_0)}{x_0(M-x)}\right)$$
So:
$$\frac{x}{x_0}\frac{M-x_0}{M-x}=e^{Mt}$$
Look solvable now...

Last edited: Sep 1, 2004
6. Sep 1, 2004

### Tide

Thanks - I flipped a sign!