Find and solve the variational equation $\dot x=x(1-x)$, $x(0)=\alpha$, for the given solution $\bar x(t)=1$

Question

I'm currently looking into learning more about the variational equation and trying to get a better grasp on the subject as a whole, if anyone could point me in the right direction for problems of this nature it would be greatly appreciated

Answer 1

I presume you have a typo for $\bar x$ and instead meant $\dot x$.

Solving via separation of variables, using partial fractions, and integrating gives:

$$t + C = \ln\left|\frac{x}{1-x}\right|$$

Where $C$ is a constant of integration, and $x \notin\{0,1\}$.

Using the boundary conditions, the general solution is:

$$\frac{x}{1-x}=\frac{\alpha}{1-\alpha}e^t$$

Hence you can now study this simpler equation.