Consider the differential equation y 6y 12y8y 0 defined o

Consider the differential equation y\"\' - 6y\" + 12y,-8y = 0 defined on the interval (-infinity, infinity). Verify that each of the functions, e^2x, xe^2x, and x^2e^2x, is a solution of this differential equation on (-infinity, infinity). Verify that these functions, e^2x, xe^2x, and x^2e^2x, form a linearly independent set on (-infinity, infinity), using the Wronskian. Find the general solution of this differential equation.

Solution