Question 1
a) Find the general solution to $y^{\prime \prime}+3 y^{\prime}+2 y=48 e^{2 x}$
\[
y(x)=
\]
(use $c_{1}$ and $c_{2}$ for your constants)

Step 1 ：The given differential equation is a non-homogeneous second order linear differential equation. The general solution of such an equation is the sum of the general solution of the corresponding homogeneous equation and a particular solution of the non-homogeneous equation.

Step 2 ：The homogeneous equation corresponding to the given differential equation is \(y'' + 3y' + 2y = 0\). The roots of the characteristic equation of this homogeneous equation, \(r^2 + 3r + 2 = 0\), will give us the general solution of the homogeneous equation.

Step 3 ：The roots of the characteristic equation are -2 and -1. Therefore, the general solution of the homogeneous equation is \(y_h(x) = c_1 e^{-2x} + c_2 e^{-x}\).

Step 4 ：The particular solution of the non-homogeneous equation can be found using the method of undetermined coefficients. Since the right hand side of the given equation is of the form \(e^{2x}\), we assume a particular solution of the form \(Ae^{2x}\) and substitute it into the equation to solve for A.

Step 5 ：The coefficient A for the particular solution is 4. Therefore, the particular solution of the non-homogeneous equation is \(y_p(x) = 4 e^{2x}\).

Step 6 ：The general solution of the given differential equation is the sum of the general solution of the homogeneous equation and the particular solution of the non-homogeneous equation.

Step 7 ：\(\boxed{y(x) = c_1 e^{-2x} + c_2 e^{-x} + 4 e^{2x}}\)