# how to linearize this differential equation?

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

• What is known about the constants $k_1, k_2, k_3, k_4$? You will need at least $k_2^2 - 4 k_3 k_4 \geq 0$ to have equilibria at all. Or $k_3 = 0$, or $k_4 = 0$. – SampleTime Nov 6 at 18:41
• Also, from your previous question, it seems you are looking for a transfer function... but that would require an input because transfer functions describe input-output relationships. However, your ODE does not have an input? – SampleTime Nov 6 at 19:17
• k1 = 3000000, k2 = 98060, k3 =1.527e+13, k4 =-19.254, k2^2-4*k3*k4 = 1.1767e+15. Is it necessary to put the input to linearize? I would put the input after getting the transfer function of the output, so I would put the input only on simulink in matlab. – Rhandrey Maestri Nov 6 at 19:35
• in fact, can I linearize the component that had the non-linear factor before working on the equation? it came from: $\Delta P_{f} = kv^2$ being the constant k, known – Rhandrey Maestri Nov 6 at 20:08
• You don't need to specify how the input "looks like". But you need an input in your ODE, otherwise the concept of transfer functions doesn't make sense because your system then doesn't have an input. However, the fact that you want to somehow apply the input "afterwards" suggests that your ODE is wrong, i.e. is lacking the input variable. – SampleTime Nov 6 at 20:11

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.