$\displaystyle \dot{x}=x-x^2-\lambda$ where $\displaystyle \lambda \ge 0$ is a constant.
It's not Bernoulli, since it's not of the form
$\displaystyle \dot{x}= ax + bx^n$
So what is it? And how do I solve it?
The Differential equation is a special equation called the Riccati equation. It reduces to the Bernoulli's equation for $\displaystyle \lambda = 0$
The solution of this equation is not very simple. One method is to make the substitution $\displaystyle x = u(t) + y$
where u(t) is a particular solution that is already known,
y is a new dependent variable.
This substitution reduces the equation to a Bernoulli's equation in y and t, and you can obtain y as a function of t. Returning to the original dependent variable gives you the actual solution as $\displaystyle x(t) = u(t) + y$
In this case, you can take the particular solution to be one of the roots $\displaystyle \alpha$ of the equation $\displaystyle x^{2}-x+\lambda = 0$ wherever $\displaystyle \alpha$ is real.
The solution will depend on $\displaystyle \lambda$ since $\displaystyle x^2 - x + \lambda$ might have:
Case 1: two distinct linear factors,
Case 2: one repeated linear factor, or
Case 3. no real linear factor
depending on the value of $\displaystyle \lambda$. For case 1, use partial fractions. For case 2, it's a standard form. For case 3 you need to complete the square and then recognise a standard form that will give you arctan.
The work hack work is left for you. (You might find it easier to start with concrete values for $\displaystyle \lambda$ for each case. Note that there is only one value for case 2).