Sorry for the bad title. I didn't know how to call it.
1. Motivation. To show the motivation of the question more explicitly, I will use an example of damped oscillation which occurs very commonly in physics. From the first equation of the below, we see that the LHS is (elastic force) + (air resistance modelled as being proportional to velocity), and the RHS is the net force according to Newton's second law.
$$\sum F_x = -kx - bv_x = ma_x$$
$$\therefore -kx -b \frac{dx}{dt} = m \frac{d^2x}{dt^2}$$
We can make the second equation look better by putting everything to one side and dividing by $m$.
$$x'' + \frac{b}{m} x' + \frac{k}{m} x=0$$
With the ansatz set as $x(t) = e^{\lambda t}$, we get $$\lambda ^2 + \frac{b}{m} \lambda + \frac{k}{m} = 0$$ and therefore
$$\lambda = \frac{-\frac{b}{m} \pm \sqrt{(\frac{b}{m})^2 - 4\frac{k}{m}}}{2} = -\frac{b}{2m} \pm \sqrt{(\frac{b}{2m})^2 - \frac{k}{m}} $$
2. My speculation.
I thought the solution will be such as
$$x(t) = e^{-(b/2m)t}(Ae^{\tilde{\omega} t} + Be^{-\tilde{\omega} t})$$, where $A, B$ are arbitrary constants to be determined and $$\tilde{\omega} = \sqrt{(\frac{b}{2m})^2 - \frac{k}{m}}$$
3. What's in the Textbooks.
I checked several textbooks, such as Serway, et al., Physics for Scientists and Engineers and Kreyszig, et al., Advanced Engineering Mathematics, and I found that all of them use another form of the differential equation.
So, they instead defined the $\lambda$ to be as such, intentionally inducing the $i$ in order to allow for a solution in the sinusoidal form through the use of Euler's formula and of the fact that linear combination of solutions is also a solution of an ODE.
$$\lambda = -\frac{b}{2m} \pm \sqrt{(\frac{b}{2m})^2 - \frac{k}{m}} = -\frac{b}{2m} \pm \sqrt{\frac{k}{m} -(\frac{b}{2m})^2 }\ \ i$$
And they used the following form of a solution:
$$x(t) = e^{-(b/2m)t} (A\cos\omega t + B\sin\omega t)$$
where $\omega$ is appropriately defined as $$\omega = \sqrt{\frac{k}{m} -(\frac{b}{2m})^2 }$$
Yes, I understand how the identities $\cos \theta = (e^{iwt} + e^{-iwt})/2$ and $\sin \theta = (e^{iwt} - e^{-iwt})/2i$ are used. However, I don't really get how we can say that $$Ae^{\tilde{\omega} t} + Be^{-\tilde{\omega} t}$$
and
$$A\cos\omega t + B\sin\omega t$$
are physically identical. They don't seem to be the models of an identical phenomenon.