I want to know the time-step $\Delta t$ for which the explicit Euler method is stable.
$$\frac{dx}{dt} = \mu + \lambda x.$$
I know for $\mu = 0$ that we need $\Delta t \leq \frac{2}{|\lambda|}$ where $\lambda$ is purely real. But now I want to include a constant $\mu$ (again purely real). How can I find my restriction for $\Delta t$?
I started with applying explicit Euler: \begin{align*} x_{i+1} &= x_i + \Delta t (\mu + \lambda x_i)\\ &= x_i(1+\Delta t \lambda) + \Delta t\mu \end{align*} The next step would be to apply that $|\frac{x_{i+1}}{x_i}|\leq 1$, however it is not possible to write $x_{i+1} = \nu x_i$ for some $\nu$, how do I have to approach this?