Solve the differential equation $y^{\mathrm{\text{'}\text{'}}}+2y^{\mathrm{\text{'}}}=2x+9-e^{-2x}$ using the method of undetermined coefficients. Which of the following is a particular solution?

• A. $y_p=2x+9+e^{-2x}$
• B. $y_p=x^2+9x-e^{-2x}$
• C. $y_p=x+\frac{9}{2}-\frac{1}{2}{e}^{-2x}$
• D. $y_p=a{x}^2+bx+c+d{e}^{-2x}$

Answer: (C)

Verified step by step guidance

1. Step 1: Identify the type of differential equation. The given equation is a second-order linear differential equation with constant coefficients: y'' + 2y' = 2x + 9 - e^{-2x}. The right-hand side is a combination of polynomial, constant, and exponential terms, which suggests using the method of undetermined coefficients.
2. Step 2: Analyze the form of the particular solution. The method of undetermined coefficients involves guessing a form for the particular solution based on the right-hand side of the equation. For the polynomial term 2x + 9, we guess a solution of the form ax + b. For the exponential term -e^{-2x}, we guess a solution of the form ce^{-2x}. Combine these guesses into a single form: y_p = ax + b + ce^{-2x}.
3. Step 3: Substitute the guessed particular solution into the differential equation. Compute the first derivative y_p' and second derivative y_p'' of the guessed solution y_p = ax + b + ce^{-2x}. Substitute y_p, y_p', and y_p'' into the original equation y'' + 2y' = 2x + 9 - e^{-2x}.
4. Step 4: Solve for the coefficients a, b, and c. After substitution, equate the coefficients of like terms (polynomial, constant, and exponential) on both sides of the equation. This will yield a system of equations for a, b, and c. Solve this system to determine the values of a, b, and c.
5. Step 5: Write the particular solution. Once the coefficients a, b, and c are determined, substitute them back into the guessed form y_p = ax + b + ce^{-2x} to obtain the particular solution. Verify that this solution satisfies the original differential equation.