Y(n)=2y(n-1)-0.5y(n-2)+x(n)-2x(n-1)+x(n-2)

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Solution for Y(n)=2y(n-1)-0.5y(n-2)+x(n)-2x(n-1)+x(n-2) equation: 


Simplifying
Y(n) = 2y(n + -1) + -0.5y(n + -2) + x(n) + -2x(n + -1) + x(n + -2)

Multiply Y * n
nY = 2y(n + -1) + -0.5y(n + -2) + x(n) + -2x(n + -1) + x(n + -2)

Reorder the terms:
nY = 2y(-1 + n) + -0.5y(n + -2) + x(n) + -2x(n + -1) + x(n + -2)
nY = (-1 * 2y + n * 2y) + -0.5y(n + -2) + x(n) + -2x(n + -1) + x(n + -2)

Reorder the terms:
nY = (2ny + -2y) + -0.5y(n + -2) + x(n) + -2x(n + -1) + x(n + -2)
nY = (2ny + -2y) + -0.5y(n + -2) + x(n) + -2x(n + -1) + x(n + -2)

Reorder the terms:
nY = 2ny + -2y + -0.5y(-2 + n) + x(n) + -2x(n + -1) + x(n + -2)
nY = 2ny + -2y + (-2 * -0.5y + n * -0.5y) + x(n) + -2x(n + -1) + x(n + -2)

Reorder the terms:
nY = 2ny + -2y + (-0.5ny + 1y) + x(n) + -2x(n + -1) + x(n + -2)
nY = 2ny + -2y + (-0.5ny + 1y) + x(n) + -2x(n + -1) + x(n + -2)

Multiply x * n
nY = 2ny + -2y + -0.5ny + 1y + nx + -2x(n + -1) + x(n + -2)

Reorder the terms:
nY = 2ny + -2y + -0.5ny + 1y + nx + -2x(-1 + n) + x(n + -2)
nY = 2ny + -2y + -0.5ny + 1y + nx + (-1 * -2x + n * -2x) + x(n + -2)

Reorder the terms:
nY = 2ny + -2y + -0.5ny + 1y + nx + (-2nx + 2x) + x(n + -2)
nY = 2ny + -2y + -0.5ny + 1y + nx + (-2nx + 2x) + x(n + -2)

Reorder the terms:
nY = 2ny + -2y + -0.5ny + 1y + nx + -2nx + 2x + x(-2 + n)
nY = 2ny + -2y + -0.5ny + 1y + nx + -2nx + 2x + (-2 * x + n * x)

Reorder the terms:
nY = 2ny + -2y + -0.5ny + 1y + nx + -2nx + 2x + (nx + -2x)
nY = 2ny + -2y + -0.5ny + 1y + nx + -2nx + 2x + (nx + -2x)

Reorder the terms:
nY = nx + -2nx + nx + 2ny + -0.5ny + 2x + -2x + -2y + 1y

Combine like terms: nx + -2nx = -1nx
nY = -1nx + nx + 2ny + -0.5ny + 2x + -2x + -2y + 1y

Combine like terms: -1nx + nx = 0
nY = 0 + 2ny + -0.5ny + 2x + -2x + -2y + 1y
nY = 2ny + -0.5ny + 2x + -2x + -2y + 1y

Combine like terms: 2ny + -0.5ny = 1.5ny
nY = 1.5ny + 2x + -2x + -2y + 1y

Combine like terms: 2x + -2x = 0
nY = 1.5ny + 0 + -2y + 1y
nY = 1.5ny + -2y + 1y

Combine like terms: -2y + 1y = -1y
nY = 1.5ny + -1y

Solving
nY = 1.5ny + -1y

Solving for variable 'n'.

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

Add '-1.5ny' to each side of the equation.
nY + -1.5ny = 1.5ny + -1.5ny + -1y

Combine like terms: 1.5ny + -1.5ny = 0.0
nY + -1.5ny = 0.0 + -1y
nY + -1.5ny = -1y

Combine like terms: -1y + y = 0
nY + -1.5ny + y = 0

The solution to this equation could not be determined.

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