I am trying to determine the order of the following partial differential equations and then trying to determine if they are linear or not, and if not why?
a) $$x^2 \frac{\partial^2u}{\partial x^2} - \left(\frac{\partial u}{\partial x}\right)^2 + x^2 \frac{\partial^2u}{\partial x \partial y } - 4 \frac{\partial^2u}{\partial y^2} = 0 $$
For a) the order would be 2 since its the highest partial derivative, and I believe its non linear because the dependent variable, $u$ (and its derivatives) appear in terms with degree that is not $1$ since the second term is squared.
b)
$$8x \frac{\partial u}{\partial y} - \frac{\partial u}{\partial x}\frac{\partial u}{\partial y}-2e^{xy}=0$$
For b) I think the order is $1$ and it is linear but I am not sure due to the second term.
Looking for some help with these two examples, thanks!
Both of them are non-linear PDE: the first one because of $\large{(\frac{du}{dx})^2}$, the second one because of $\frac{du}{dx}\frac{du}{dy}$. In linear PDE the coefficients of the function and its derivatives cannot depend on function itself or any of its derivatives.