y(t)=x(t-1)+x(t+1)

Solution for y(t)=x(t-1)+x(t+1) equation:

Simplifying
y(t) = x(t + -1) + x(t + 1)

Multiply y * t
ty = x(t + -1) + x(t + 1)

Reorder the terms:
ty = x(-1 + t) + x(t + 1)
ty = (-1 * x + t * x) + x(t + 1)

Reorder the terms:
ty = (tx + -1x) + x(t + 1)
ty = (tx + -1x) + x(t + 1)

Reorder the terms:
ty = tx + -1x + x(1 + t)
ty = tx + -1x + (1 * x + t * x)

Reorder the terms:
ty = tx + -1x + (tx + 1x)
ty = tx + -1x + (tx + 1x)

Reorder the terms:
ty = tx + tx + -1x + 1x

Combine like terms: tx + tx = 2tx
ty = 2tx + -1x + 1x

Combine like terms: -1x + 1x = 0
ty = 2tx + 0
ty = 2tx

Solving
ty = 2tx

Solving for variable 't'.

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

Add '-2tx' to each side of the equation.
-2tx + ty = 2tx + -2tx

Combine like terms: 2tx + -2tx = 0
-2tx + ty = 0

Factor out the Greatest Common Factor (GCF), 't'.
t(-2x + y) = 0

Subproblem 1
Set the factor 't' equal to zero and attempt to solve:

Simplifying
t = 0

Solving
t = 0

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

Simplifying
t = 0
Subproblem 2
Set the factor '(-2x + y)' equal to zero and attempt to solve:

Simplifying
-2x + y = 0

Solving
-2x + y = 0

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

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

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

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

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

Simplifying
0 = 2x + -1y

The solution to this equation could not be determined.

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