Consider the equation $u_{xx}+2u_{xy}+u_{yy}=0.$ Write the equation in the coordinates $s=x, \ t =x-y$ and find the general solution of the equation.
So we have that $x=s$ and $y=s-t$ then assume that $u$ is a solution of the pde and thus have $v(s,t)=u(s,s-t)$. But a hint gives that $v_s=u_x+u_y$ and $v_{ss}=u_{xx}+u_{xy}+u_{yx}+u_{yy},$ how was that obtained?
For any function $z(x,y)$, the chain rule gives $$\frac{\partial z}{\partial s}=z_xx_s+z_yy_s=z_x+z_y\ .\tag{$*$}$$ Applying this to $z=u$ gives $$\frac{\partial u}{\partial s}=u_x+u_y\ ,$$ and so $$\frac{\partial^2u}{\partial s^2} =\frac{\partial u_x}{\partial s}+\frac{\partial u_y}{\partial s}\ .\tag{$*\!*$}$$ But now applying $(*)$ to $u_x$ and $u_y$ gives $$\frac{\partial u_x}{\partial s}=u_{xx}+u_{xy}\ ,\quad \frac{\partial u_y}{\partial s}=u_{yx}+u_{yy}\ ;$$ substituting back into $(**)$ gives $$u_{ss}=u_{xx}+2u_{xy}+u_{yy}\ ,$$ assuming that $u_{yx}=u_{xy}$.
Writing this in terms of a different function $v$ is probably, strictly speaking, correct. But IMHO it is really just confusing, and I prefer to think of it as the same quantity, just written in terms of different variables. By way of comparison, think of writing the circumference of a circle as $$C=2\pi r=u(r)\ ,\quad C=\pi d=v(d)\ .$$ These are two different functions, but surely the easy way to think of it is that it is the same quantity written in terms of different variables.