User:Egm6341.s11.team4.kurth/hw7

Exact Solution
Starting with the logistic equation and separating the variables, it becomes


 * $$\displaystyle \frac{1}{x(1-\frac{x}{x_{max}})}\;dx=r\;dt$$

Now, multiply the left-hand side of the equation by $$\frac{x_{max}}{x_{max}}$$  to simplify the denominator


 * $$\displaystyle \frac{x_{max}}{x(x_{max}-x)}\;dx=r\;dt$$

The left-hand side can now be deconstructed into the partial fractions


 * $$\displaystyle \left[\frac{1}{x}+\frac{1}{x_{max}-x}\right]dx=r\;dt$$

Integrating both sides yields


 * $$\displaystyle ln|x|-ln|x_{max}-x|-ln|x_0|+ln|x_{max}-x_0|=r(t-t_0)$$

Taking the inverse natural log of both sides


 * $$\displaystyle \frac{x(x_{max}-x_0)}{x_0(x_{max}-x)}=e^{r(t-t_0)}$$

Now algebraically solving for x


 * $$\displaystyle x(x_{max}-x_0)=x_0(x_{max}-x)e^{r(t-t_0)}$$


 * $$\displaystyle x(x_{max}-x_0)=x_0x_{max}e^{r(t-t_0)}-x_0xe^{r(t-t_0)}$$


 * $$\displaystyle x(x_{max}-x_0+x_0e^{r(t-t_0)})=x_0x_{max}e^{r(t-t_0)}$$


 * $$\displaystyle x = \frac{x_0x_{max}e^{r(t-t_0)}}{x_{max}-x_0+x_0e^{(t-t_0)}}$$