Given a function $f\in C^1(\mathbb{R}^2,\mathbb{R})$, consider the following system of differential equations: $$ \frac{\partial u}{\partial x}(x,y) = \frac{\partial f}{\partial x}(x,y) \quad,\quad \frac{\partial u}{\partial y}(x,y) = \alpha\,\frac{\partial f}{\partial y}(x,y) $$ for a given $\alpha>0$. What are all possible solutions $u\in C^1(\mathbb{R}^2,\mathbb{R})$ ?
The followings are not solutions: $$ u(x,y)=f(x,\alpha\,y)+C $$ or $$ u(x,y)=\alpha\,f\left(\frac{1}{\alpha}\,x,y\right)+C \;.$$ Thanks @TedShifrin
If you assume $f\in C^2$ and $\alpha\ne 1$, you deduce that $f_{xy}=0$, from which you conclude that $f(x,y) = g(x)+h(y)$ for some $C^1$ functions $g$ and $h$. Then $u(x,y)= g(x)+\alpha h(y)$ gives the general solution (you can absorb the constant).
Now, can you show that $f$ takes this form without assuming $f\in C^2$?
What happens if $\alpha = 1$?