In search for analytical solutions for sixth order nonlinear PDE Explaining a bubble's motion.

Question

I am modelling the nonlinear behaviour of an bubble in hot water. I am trying to explain it's rotational, vibrational and translational motion in water with impurities and subject to varying temperatures. I got the following equation. How do I solve the following equation analytically?

$$x^2\frac{∂^6x}{∂y^6}+\frac32(\frac{∂^2x}{∂y^2})^{2}+x\frac{∂x}{∂z}-\frac{Ax}{\frac{∂^3x}{∂z^3}}+B\frac{∂⁵x}{∂z⁵}+\frac{x^{3}e^{sinhx}}{∂²x/∂y²}+\frac{7}{11}\frac{e^{\frac{x⁴}{sinhx}}}{x^{3}}\frac{∂³x}{∂y³}=C$$
Where $x=x(z,y)$ and $A$,$B$ and C are constants. I would also appreciate if the graph is provided.

Comments

Could you please derive your PDE?