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?
$$x^2\frac{\partial^6x}{\partial y^6}+\frac32\left(\frac{\partial^2x}{\partial y^2}\right)^{2}+x\frac{\partial x}{\partial z}-\frac{Ax}{{\partial ^3x/\partial z^3}}+B~\frac{\partial^5 x}{\partial z^5 }+\frac{x^{3}e^{\operatorname{sinh}x}}{\partial^2 x/\partial y^2}+\frac{7}{11}\frac{e^{\frac{x^4}{\operatorname{sinh}x}}}{x^{3}}\frac{\partial^3x}{\partial y^3}=C$$
Where $x=x(z,y)$ and $A$,$B$ and $C$ are constants. I would also appreciate if the graph is provided.