Solve the differential equation using variation of parameters: $y^{\mathrm{\text{'}\text{'}}}+3y^{\mathrm{\text{'}}}+2y=17+e^x$. Which of the following is the general solution?

• A. $y=C_1{e}^{-x}+C_2{e}^{-2x}+17+e^x$
• B. $y=C_1{e}^{-x}+C_2{e}^{-2x}+17x+x{e}^x$
• C. $y=C_1{e}^{2x}+C_2{e}^{-3x}+17x+e^x$
• D. $y=C_1{e}^{-x}+C_2{e}^{-2x}+\frac{17}{2}+\frac{1}{6}{e}^x$

Answer: (D)

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 = 17 + e^x.
2. Step 2: Solve the corresponding homogeneous equation. The homogeneous equation is y'' + 3y' + 2y = 0. Solve this by finding the characteristic equation: r^2 + 3r + 2 = 0. Factorize the characteristic equation to find the roots, which will determine the complementary solution.
3. Step 3: Write the complementary solution. Based on the roots of the characteristic equation (r = -1 and r = -2), the complementary solution is y_c = C_1 e^{-x} + C_2 e^{-2x}, where C_1 and C_2 are constants.
4. Step 4: Use variation of parameters to find a particular solution. For the non-homogeneous term (17 + e^x), apply the method of variation of parameters. This involves finding two functions u_1(x) and u_2(x) that satisfy specific integrals derived from the complementary solution and the non-homogeneous term.
5. Step 5: Combine the complementary and particular solutions. The general solution is the sum of the complementary solution and the particular solution. After solving for u_1(x) and u_2(x), substitute them back to find the particular solution. The final general solution will be y = C_1 e^{-x} + C_2 e^{-2x} + (17/2) + (1/6)e^x.