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# Functional Equations

Functional equations are equations where the unknowns are the functions. Rather than solving for x, you solve for the function in questions like "Find all functions that have these properties."

The function \(f \) satisfies \[(x-y)f(x+y)-(x+y)f(x-y)+4xy(x^2-y^2)=0,\]

whenever \( |x| \neq |y | \).

Given that \(f(1)=2,\) what is \(f(x)\)?

A function \(f\) from the positive integers to the positive integers satisfies the conditions

1) \(f(n) = 2 f(n-1) + 3 \),

2) \( f(1) = 9 .\)

What is the value of \( f(9) \)?

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