Final answer:
To find the appropriate form of the particular solution for the given differential equation, we can use the method of undetermined coefficients. Assuming the particular solution has the form (Ax^2 + Bx + C)e^(-x), we can solve for the coefficients by substituting this form into the equation and equating the coefficients of the like terms. The appropriate form of the particular solution is then (Ax^2 + Bx + C)e^(-x).
Step-by-step explanation:
The particular solution for the given differential equation can be found by using the method of undetermined coefficients. Since the right-hand side of the equation is a product of a polynomial (4x^2 - 5) and an exponential function (e^(-x)), we assume that the particular solution has the same form: yp(x) = (Ax^2 + Bx + C)e^(-x), where A, B, and C are coefficients to be determined.
Substituting this assumed form into the differential equation, we can solve for the coefficients. Differentiating yp(x) twice and substituting it back into the equation, we get 4Ae^(-x) - 2Ae^(-x) - 2(Ax^2 + Bx + C)e^(-x) = (4x^2 - 5)e^(-x).
By equating the coefficients of the like terms on both sides of the equation, we can obtain a system of equations to solve for A, B, and C. Once the coefficients are found, the appropriate form of the particular solution yp(x) is given by yp(x) = (Ax^2 + Bx + C)e^(-x).