This formula (trick) is directly given in my study material. I have tried to prove it but its getting too long.Please help by giving proof of this condition, ie,formula.Thanks in advance.
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If $y$ is a function of $x$, then differentiating with respect to $x$ gives $$f'(y/x)\left(\frac{dy}{dx}\frac{1}{x}-\frac{y}{x^{2}}\right)=0$$ Then, if $f'(y/x) \ne 0$, $$\frac{dy}{dx}=\frac{y}{x}$$
preferred_anon
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If $f\left(\dfrac x y\right)$ is constant then $\dfrac x y$ is constant (or is any of several constants in the inverse-image under $f$ of the constant value of $f(x/y)$.
