A region's mouse population in year n is given by the second-order difference equation Pn=2Pn-1+8Pn-2. If A and B are constants that depend on current and past populations, what general form must an equation for the population in year n take?

Pn = A * e ^ (4 * n) + B * e ^ ( - 2 * n)

Step-by-step explanation:

We have the following:

Pn = 2 * P_n-1 + 8P_n-2

the characteristic equation is as follows:

L ^ 2 - 2 * L - 8 = 0

Which is equal to:

L ^ 2 - 4 * L + 2 * L - 8 = 0

we factor:

L * (L - 4) + 2 * (L - 4) = 0

From here we can deduce that:

L = - 2

L = 4

Thus:

Pn = A * e ^ (4 * n) + B * e ^ ( - 2 * n)