Solve the differential equation $y^{\mathrm{\text{'}\text{'}}}+2y=-18x^2{e}^{2x}$ using the method of undetermined coefficients. Which of the following is a particular solution?

• A. $y_p\left(x\right)=(-3x^2+6x-9)e^{2x}$
• B. $y_p\left(x\right)=(3x^2+3x+3)e^{2x}$
• C. $y_p\left(x\right)=(-9x^2+6x-3)e^{2x}$
• D. $y_p\left(x\right)=(3x^2-6x+9)e^{2x}$

Answer: (A)

Verified step by step guidance

1. Step 1: Recognize that the differential equation y'' + 2y = -18x^2 e^{2x} is non-homogeneous. The method of undetermined coefficients is suitable because the right-hand side (-18x^2 e^{2x}) is a product of a polynomial and an exponential function.
2. Step 2: Identify the form of the particular solution y_p(x). Since the right-hand side is -18x^2 e^{2x}, propose a solution of the form y_p(x) = (Ax^2 + Bx + C)e^{2x}, where A, B, and C are constants to be determined.
3. Step 3: Compute the derivatives of y_p(x). First, find y_p'(x) and y_p''(x): y_p'(x) = [(2Ax^2 + 2Bx + 2C) + (2Ax + B)]e^{2x}, and y_p''(x) = [(4Ax^2 + 4Bx + 4C) + (4Ax + 2B) + (2A)]e^{2x}.
4. Step 4: Substitute y_p(x), y_p'(x), and y_p''(x) into the original differential equation y'' + 2y = -18x^2 e^{2x}. Combine like terms and equate coefficients of x^2, x, and the constant term to solve for A, B, and C.
5. Step 5: Solve the resulting system of equations for A, B, and C. Once the values of A, B, and C are determined, substitute them back into y_p(x) = (Ax^2 + Bx + C)e^{2x} to find the particular solution.