Equation of motion of harmonic oscillator with dissipation function of speed

We have a mass sustained by a spring $K$ and a damper $C$, with a base excitation. Let’s call $s(t)$ the base excitation and $x(t)$ the mass motion.
The differential equation of this system will be:

$m cdot ddot{x} + C cdot (dot{x(t)} – dot{s(t)}) + K cdot (x(t) – s(t)) = 0$

Suppose that $C$ is not a constant, but:

$C = C_1/(dot{x(t)} – dot{s(t)})$

And so the viscous force will be:

$F_v = -C_1$

Using the newtonian formulation to obtain the differential equation, I obtain:

$m cdot ddot{x} + K cdot (x(t) – s(t)) + C_1 = 0$

But if I want to use Lagrangian formulation, by adding the Rayleigh dissipation function:

$D = 1/2 cdot C cdot (dot{x(t)} – dot{s(t)})^2 = 1/2 cdot C_1 cdot (dot{x(t)} – dot{s(t)})$

I obtain the differential equation:

$m cdot ddot{x} + K cdot (x(t) – s(t)) + C_1/2 = 0$

That’s not the same! Which one is correct? And why this difference?