Question

Which of the following functions are solutions of the differential equation y”– 4y’+4y = e

x?

(a) e^x

(b) e^2x

(c) e^2x + e^x

(d) xe^2x + e^x

(e) e^2x + xe^x

Answer 1

$\text{Using the method of undetermined co-efficients we find the particular solution to the}\\ \text{given differential equation.Let $y_p$ represent the particualar solution. }\\ \therefore y_p = Ae^t \text{ where A is a constant}\\ \implies y_p' = Ae^t, y_p'' = Ae^t\\ \text{Substituting into the given differential equation we have that}\\ Ae^t-4Ae^t+4Ae^t=e^t\\ \implies Ae^t = e^t\\ \therefore A = 1\\ \text{Hence, $e^t$ is a solution to the given differential equation.}$