Finding a function satisfying the given differential equation.

Let $varphi(x) := f(x) fleft(frac{a}{x}right)$. Differentiating we get, for every $x > 0$, $$ varphi'(x) = f'(x) fleft(frac{a}{x}right)+f(x) left[fleft(frac{a}{x}right)right]' = frac{a}{x fleft(frac{a}{x}right)} fleft(frac{a}{x}right)- frac{a}{x^2} f(x) frac{x}{f(x)} = 0. $$ Hence there exists $c > 0$ such that $varphi(x) = c$ for every $x > 0$. Since $$ f'(x) = frac{a}{x f(a/x)} = frac{a}{c} cdot frac{f(x)}{x} $$ integrating we get $$ f(x) = f(1) x^{a/c}. $$ Substituting in $f'(x) = frac{a}{c} f(1) x^{a/c - 1}$ the values $x=1$ and $x=2$ we finally get $a/c = 2$ and $f(1) = 1/2$, so that $a=2$ and $f(x) = x^2/2$.