TITLE: Solving for a function
QUESTION [5 upvotes]: How can I find a general solution to following equation,
$$
f\left(\frac{1}{y}\right)=y^2 f(y).
$$
I know that $f(y) = \frac{1}{1 + y^2}$ is a solution but are there more? Is there a general technique that I can read up about for problems of this kind?

REPLY [2 votes]: Let $g(x)$ be any even function. Then $f(x)=\frac{1}{x}g(\ln|x|)$ and satisfies given equation.
Proof:
$$f\left(\frac{1}{y}\right)=y^2f(y),\quad y\not=0$$
$$\frac{1}{1/y}g\left(\ln\left|\frac{1}{y}\right|\right)=y^2\frac{1}{y}g(\ln|y|)$$
$$yg(-\ln|y|)=yg(\ln|y|)$$
$$g(-\ln|y|)=g(\ln|y|),\quad \ln|y|=z$$
$$g(-z)=g(z)$$
Last equation is true, because $g(x)$ is even.