Let us have the following equation of motion (it might not necessarily correspond to a physical system):
$$\dot{x} + a \cdot x + b \cdot x^2 + c=0.$$
I would like to deduce the corresponding Lagrangian. I have tried multiple combinations but I have not been able to obtain the $\dot{x}$ term.
All the systems I have worked with before had a $\dot{x}^2$ term instead of $\dot{x}$. I have unsuccessfully tried to compare the problem with a damped harmonic oscillator, but the $\dot{x}^2$ term is still missing.
I have solved the equation just in case:
$$x=\frac{A_1 \cdot e^{A_2 t} + A_3}{1 - A_4 \cdot e^{A_2 t}}.$$
