f(x+y)=yf(x)+xf(y)

Solution for f(x+y)=yf(x)+xf(y) equation:

Simplifying
f(x + y) = yf(x) + xf(y)
(x * f + y * f) = yf(x) + xf(y)
(fx + fy) = yf(x) + xf(y)

Multiply fy * x
fx + fy = fxy + xf(y)

Multiply fx * y
fx + fy = fxy + fxy

Combine like terms: fxy + fxy = 2fxy
fx + fy = 2fxy

Solving
fx + fy = 2fxy

Solving for variable 'f'.

Move all terms containing f to the left, all other terms to the right.

Add '-2fxy' to each side of the equation.
fx + -2fxy + fy = 2fxy + -2fxy

Combine like terms: 2fxy + -2fxy = 0
fx + -2fxy + fy = 0

Factor out the Greatest Common Factor (GCF), 'f'.
f(x + -2xy + y) = 0

Subproblem 1
Set the factor 'f' equal to zero and attempt to solve:

Simplifying
f = 0

Solving
f = 0

Move all terms containing f to the left, all other terms to the right.

Simplifying
f = 0
Subproblem 2
Set the factor '(x + -2xy + y)' equal to zero and attempt to solve:

Simplifying
x + -2xy + y = 0

Solving
x + -2xy + y = 0

Move all terms containing f to the left, all other terms to the right.

Add '-1x' to each side of the equation.
x + -2xy + -1x + y = 0 + -1x

Reorder the terms:
x + -1x + -2xy + y = 0 + -1x

Combine like terms: x + -1x = 0
0 + -2xy + y = 0 + -1x
-2xy + y = 0 + -1x
Remove the zero:
-2xy + y = -1x

Add '2xy' to each side of the equation.
-2xy + 2xy + y = -1x + 2xy

Combine like terms: -2xy + 2xy = 0
0 + y = -1x + 2xy
y = -1x + 2xy

Add '-1y' to each side of the equation.
y + -1y = -1x + 2xy + -1y

Combine like terms: y + -1y = 0
0 = -1x + 2xy + -1y

Simplifying
0 = -1x + 2xy + -1y

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
f = {0}