Find all functions $F(x,y)$ such as $\frac{\sqrt{3}}{2}\frac{\partial{f}}{\partial{x}}+\frac{1}{2}\frac{\partial{f}}{\partial{y}}=0$
Solution 1:
Let´s use the method of characteristics. The equation can be converted to: $$\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\cdot \mathrm{grad}f=0$$ That means $f$ is constant along lines which direction is parallel to $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$. Hence we can express those lines by: $$x-\sqrt{3}y=k$$ For each value of $k$, $f$ assumes a different one. Hence $f$ depends on $k$, in other words $f$ is a function of $k$.
Therefore: $$f(x,y)=g(x-\sqrt{3}y)$$