Y(n)=ny(n-1)+x(n)

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Solution for Y(n)=ny(n-1)+x(n) equation: 


Simplifying
Y(n) = ny(n + -1) + x(n)

Multiply Y * n
nY = ny(n + -1) + x(n)

Reorder the terms:
nY = ny(-1 + n) + x(n)
nY = (-1 * ny + n * ny) + x(n)
nY = (-1ny + n2y) + x(n)

Multiply x * n
nY = -1ny + n2y + nx

Reorder the terms:
nY = nx + -1ny + n2y

Solving
nY = nx + -1ny + n2y

Solving for variable 'n'.

Reorder the terms:
nY + -1nx + ny + -1n2y = nx + -1nx + -1ny + ny + n2y + -1n2y

Combine like terms: nx + -1nx = 0
nY + -1nx + ny + -1n2y = 0 + -1ny + ny + n2y + -1n2y
nY + -1nx + ny + -1n2y = -1ny + ny + n2y + -1n2y

Combine like terms: -1ny + ny = 0
nY + -1nx + ny + -1n2y = 0 + n2y + -1n2y
nY + -1nx + ny + -1n2y = n2y + -1n2y

Combine like terms: n2y + -1n2y = 0
nY + -1nx + ny + -1n2y = 0

Factor out the Greatest Common Factor (GCF), 'n'.
n(Y + -1x + y + -1ny) = 0


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

Simplifying
n = 0

Solving
n = 0

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

Simplifying
n = 0

Subproblem 2
Set the factor '(Y + -1x + y + -1ny)' equal to zero and attempt to solve:

Simplifying
Y + -1x + y + -1ny = 0

Reorder the terms:
Y + -1ny + -1x + y = 0

Solving
Y + -1ny + -1x + y = 0

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

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

Reorder the terms:
Y + -1Y + -1ny + -1x + y = 0 + -1Y

Combine like terms: Y + -1Y = 0
0 + -1ny + -1x + y = 0 + -1Y
-1ny + -1x + y = 0 + -1Y
Remove the zero:
-1ny + -1x + y = -1Y

Add 'x' to each side of the equation.
-1ny + -1x + x + y = -1Y + x

Combine like terms: -1x + x = 0
-1ny + 0 + y = -1Y + x
-1ny + y = -1Y + x

Add '-1y' to each side of the equation.
-1ny + y + -1y = -1Y + x + -1y

Combine like terms: y + -1y = 0
-1ny + 0 = -1Y + x + -1y
-1ny = -1Y + x + -1y

Divide each side by '-1y'.
n = y-1Y + -1xy-1 + 1

Simplifying
n = y-1Y + -1xy-1 + 1

Reorder the terms:
n = 1 + -1xy-1 + y-1Y
Solution
n = {0, 1 + -1xy-1 + y-1Y}

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