Final answer:
To solve the given differential equation by undetermined coefficients, assume a particular solution of the form y_p = Ax^2 + Bx + C and solve for the constants. Find the general solution by adding the particular solution to the complementary function. The complementary function is of the form y_c = c1 + c2x + c3e^(-x) + c4xe^(-x).
Step-by-step explanation:
To solve the given differential equation by undetermined coefficients, we assume a particular solution of the form y_p = Ax^2 + Bx + C, where A, B, and C are constants to be determined. Substituting this into the differential equation, we can solve for the values of A, B, and C. Once we have the particular solution, we can find the general solution by adding it to the complementary function, which is the solution to the homogeneous form of the equation.
In this case, the homogeneous form of the equation is y'''' + 2y'' + y = 0. Using the characteristic equation, we find that the complementary function is of the form y_c = c1 + c2x + c3e^(-x) + c4xe^(-x), where c1, c2, c3, and c4 are constants. Thus, the general solution is y = y_c + y_p = c1 + c2x + c3e^(-x) + c4xe^(-x) + Ax^2 + Bx + C.