Solve the differential equation. (Assume the independent variable is x.)y″ − 8y ′ + 16y = 0

The given differential equation is a second order homogeneous linear differential equation with constant coefficients. The general form of such an equation is:

ay″ + by′ + c*y = 0

In this case, a = 1, b = -8, and c = 16.

The characteristic equation of the differential equation is given by:

ar^2 + br + c = 0

Substituting the values of a, b, and c, we get:

r^2 - 8*r + 16 = 0

This can be factored to:

(r - 4)^2 = 0

Setting this equal to zero gives the roots r = 4 (double root).

The general solution of a second order homogeneous linear differential equation with constant coefficients and distinct roots is given by:

y = c1e^(r1x) + c2e^(r2x)

However, in the case of a double root (r1 = r2 = r), the general solution takes a slightly different form:

y = (c1 + c2*x)e^(rx)

Substituting r = 4, we get the general solution of the given differential equation:

y = (c1 + c2*x)e^(4x)

where c1 and c2 are arbitrary constants that can be determined from initial conditions or boundary conditions.