calculate the difference equation whose general solution is Y=(xc)^2-1 with c constant then demonstrate with each of the following functions if they are a solution to said equation and indicate what type of solution it is a) y=x^2 b) y=-1 c)y=x^2+2x

To find the difference equation for the given general solution \(Y = (xc)^2 - 1\), we need to find the second-order difference equation that this solution satisfies. A second-order difference equation relates a sequence to its past values. Let's proceed step by step. The general solution is \(Y = (xc)^2 - 1\). Now, let's find the first and second differences of \(Y\) with respect to \(x\): First difference: \(\Delta Y = Y(x + 1) - Y(x)\) \(\Delta Y = [(x + 1)c]^2 - 1 - (xc)^2 + 1\) \(\Delta Y = [c^2(x^2 + 2x + 1) - (xc)^2]\) \(\Delta Y = c^2x^2 + 2c^2x + c^2 - x^2c^2\) \(\Delta Y = 2c^2x + c^2\) Second difference: \(\Delta^2 Y = \Delta Y(x + 1) - \Delta Y(x)\) \(\Delta^2 Y = [2c^2(x + 1) + c^2] - [2c^2x + c^2]\) \(\Delta^2 Y = 2c^2x + 2c^2 + c^2 - 2c^2x - c^2\) \(\Delta^2 Y = 2c^2\) So, the second-order difference equation is \(\Delta^2 Y = 2c^2\). Now, let's test each of the given functions to see if they are solutions to this difference equation: a) \(y = x^2\): To check if \(y = x^2\) is a solution, we need to find \(\Delta^2 y\) and see if it equals \(2c^2\). \(\Delta^2 y = 2\) Since \(\Delta^2 y\) is not equal to \(2c^2\) for any constant \(c\), \(y = x^2\) is not a solution to the difference equation. b) \(y = -1\): To check if \(y = -1\) is a solution, we need to find \(\Delta^2 y\) and see if it equals \(2c^2\). \(\Delta^2 y = 0\) Since \(\Delta^2 y\) is not equal to \(2c^2\) for any constant \(c\), \(y = -1\) is not a solution to the difference equation. c) \(y = x^2 + 2x\): To check if \(y = x^2 + 2x\) is a solution, we need to find \(\Delta^2 y\) and see if it equals \(2c^2\). \(\Delta^2 y = 2\) Since \(\Delta^2 y\) is not equal to \(2c^2\) for any constant \(c\), \(y = x^2 + 2x\) is not a solution to the difference equation. None of the provided functions is a solution to the given difference equation \(\Delta^2 Y = 2c^2\).