Trying to work backward to the solution for this nonlinear ODE in a control book I have:
$\frac{\mathrm{dx} }{\mathrm{d} t}=-x+x^2$
Book lists the solution to the ODE as:
$x(t)=\frac{x_0e^-t}{1-x_0+x_0e^-t}$
I used the ODE45 function in MATLAB to generate a solution from a family of initial conditions and I get the same plot as in the book.
I run into issues when I try to analytically solve this equation through separation of variables:
$\int \frac{dx}{-x+x^2}=\int dt$
$ln\left|\frac{x+1}{x} \right|=t+C$
Add my initial conditions
$ln\left|\frac{x+1}{x} \right|=t+ln\left|\frac{x_0+1}{x_0}\right|$
Multiply both sides by e
$\frac{x+1}{x}=e^t\left ( \frac{x_0+1}{x_0} \right )$
Rearrange
$x=\frac{1}{e^t\left ( \frac{x_0+1}{x_0} \right )-1}$
I use dsolve on Matlab to try come up with a general solution and I get almost the same as I worked by hand:
However, it does not match the plot from the textbook.
Unsure where I am going wrong. Can someone review my working.