Solve the differential equation using variation of parameters: $y^{\mathrm{\text{'}\text{'}}}+3y^{\mathrm{\text{'}}}+2y=\frac{1}{3}+e^x$. Which of the following is the general solution?

• A. $y=C_1{e}^{-x}+C_2{e}^{-2x}+\frac{1}{6}+\frac{1}{3}{e}^x$
• B. $y=C_1{e}^{-x}+C_2{e}^{-2x}+\frac{1}{3}+\frac{1}{2}x{e}^x$
• C. $y=C_1{e}^{-x}+C_2{e}^{-2x}+\frac{1}{6}+\frac{1}{2}x{e}^x$
• D. $y=C_1{e}^{-x}+C_2{e}^{-2x}+\frac{1}{3}+\frac{1}{3}{e}^x$

Answer: (C)

Verified step by step guidance

1. Step 1: Identify the type of differential equation. The given equation is a second-order linear non-homogeneous differential equation: y'' + 3y' + 2y = 1/3 + e^x.
2. Step 2: Solve the corresponding homogeneous equation y'' + 3y' + 2y = 0. Find the characteristic equation: r^2 + 3r + 2 = 0. Factorize it to find the roots r = -1 and r = -2. The general solution of the homogeneous equation is y_h = C_1 e^{-x} + C_2 e^{-2x}.
3. Step 3: Use the method of variation of parameters to find a particular solution y_p for the non-homogeneous equation. The method involves finding two functions u_1(x) and u_2(x) such that y_p = u_1(x)y_1 + u_2(x)y_2, where y_1 = e^{-x} and y_2 = e^{-2x} are solutions of the homogeneous equation.
4. Step 4: Compute u_1(x) and u_2(x) using the formulas derived from variation of parameters. These formulas involve integrating expressions that include the Wronskian of y_1 and y_2, as well as the non-homogeneous term (1/3 + e^x). After integration, determine the particular solution y_p.
5. Step 5: Combine the homogeneous solution y_h and the particular solution y_p to form the general solution: y = y_h + y_p = C_1 e^{-x} + C_2 e^{-2x} + 1/6 + (1/2)x e^x.