Mathematics

Problem

The homogeneous differential equation

y''-4y'+4y=0

is given. A family of exponential functions is proposed in the form

y(x)=e^{kx}

, where

k

is a real constant. (a) Determine

y'(x)

and

y''(x)

for the proposed function. (b) Substitute these expressions into the differential equation and find all real values of

k

for which

y(x)=e^{kx}

satisfies the equation. (c) For each distinct value of

k

obtained in part (b), write an independent solution and state the dimension of the solution space for the differential equation.