Consider the nonhomogeneous Cauchy Euler's differential equation,

$y^{\text{'}\text{'}}+4y^{\text{'}}+8y=e^{2t},y(0)=0,y^{\text{'}}(0)=2$

$($a$)$ Find a complimentary solution, $y_c$ of the homogeneous differential equation.

$($b$)$ Find a particular solution, $y_p$ by using variation of parameters. Write down the general solution of the nonhomogeneous differential equation.

$($c$)$ Solve the following initial value problem using the Laplace transform method.