6. Consider the differential equation
$y'' + y' + 2y = R(x)$.
a) When $R(x) = 0$, find the general solution of the equation.
b) When $R(x) = 4x^2$, find the general solution of the equation by the method of undetermined coefficients.
c) When $R(x) = \frac{1}{e^{2x}+1}$, find a particular solution of the equation by variation of parameters.
(12 marks)
7. (a) Find the explicit solution $y(x)$ of the initial value problem
$y' + \frac{4}{x}y = \frac{e^x}{x}$, $y(1) = e$.
(b) Use the Laplace transform to solve the initial value problem
$y'' - y' - 2y = \begin{cases} 1, & 0 \le t < 1 \\ 0, & t \ge 1 \end{cases}$, $y(0) = 0$, $y'(0) = 1$.
(12 marks)
