how to linearize this differential equation?

Question

I need the transfer function of:

$K1*\dot Q(t) + K2*Q(t) + K3*Q(t)^2 + K4 = 0 $

Being K1, K2, K3 and K4 constants that depend on the system parameters. The problem is that it is a nonlinear equation, so I can not apply laplace, I tried a technique of linearization at the point of stability but resulted in 0. Any tips?

appreciate

Answer 1

Consider the differential equation $\dot{x} = f(x,t)$, suppose that there exists $x_0$ such that $f(x_0)=0$, and defive $z=x-x_0$. Then $\dot{z}\approx \frac{\partial f}{\partial x}(x_0,t)z$.

In your case, $\dot{z}(t) \approx -\frac{1}{K_1}\left(K_2 + 2K_3Q_0\right)z$, where $z(t):=Q(t)-Q_0$ and $Q_0$ is the real solution of the quadratic equation $K_3Q^2 + K_2Q + K_4 = 0$ around which you linearize.