Final answer:
To solve for the particular solution y_p(x) of the given non-homogeneous differential equation, we assume a solution with the same sine and cosine components as the non-homogeneous part, solve for the coefficients, and then express the solution in Maple syntax.
Step-by-step explanation:
To find the particular solution yp(x) of the non-homogeneous differential equation d²/dx²y(x) − 2(d/dxy(x)) + 2y(x) = 18sin(2x) - 4cos(2x), given that yh(x) = ex(A sin(x) + B cos(x)) is the general solution of the corresponding homogeneous ODE, we need to find a particular solution that fits the non-homogeneous part.
Since the right-hand side consists of sine and cosine functions with a factor of 2 in their arguments, we can expect the particular solution to have a similar form. Thus, we can guess a particular solution to be of the form yp(x) = C sin(2x) + D cos(2x). Upon substituting this guess into the original equation and differentiating, we solve for the coefficients C and D.
However, because the sine and cosine terms in the given general solution of the homogeneous equation yh(x) do not have the same frequency as those in the particular solution we are seeking, we do not run into issues concerning the terms of yp being solutions to the homogeneous part.
Once the coefficients C and D are found, the particular solution can be formulated and expressed in Maple syntax.