Solve the differential equation by variation of parameters: $y^{″}+y=\sin ⁠2x$. What is the general solution $y\left(x\right)$?

• A. $y\left(x\right)=C_1{e}^x+C_2{e}^{-x}+\frac{1}{2}\sin ⁠2x$
• B. $y\left(x\right)=C_1\cos ⁠x+C_2\sin ⁠x+\frac{1}{2}\sin ⁠2x$
• C. $y\left(x\right)=C_1\cos ⁠x+C_2\sin ⁠x+\frac{1}{3}\cos ⁠2x$
• D. $y\left(x\right)=C_1\cos ⁠x+C_2\sin ⁠x-\frac{1}{3}\sin ⁠2x$

Answer: (D)

Verified step by step guidance

1. Step 1: Identify the type of differential equation. The given equation y'' + y = sin(2x) is a second-order linear non-homogeneous differential equation. The method of variation of parameters is suitable for solving this type of equation.
2. Step 2: Solve the corresponding homogeneous equation y'' + y = 0. The characteristic equation is r^2 + 1 = 0, which has roots r = ±i. Therefore, the general solution to the homogeneous equation is y_h(x) = C_1 cos(x) + C_2 sin(x), where C_1 and C_2 are constants.
3. Step 3: Use variation of parameters to find a particular solution y_p(x) for the non-homogeneous equation. Assume y_p(x) = u_1(x) cos(x) + u_2(x) sin(x), where u_1(x) and u_2(x) are functions to be determined.
4. Step 4: Compute u_1(x) and u_2(x) using the formulas derived from variation of parameters. These involve integrating expressions that depend on the Wronskian of the solutions to the homogeneous equation and the non-homogeneous term sin(2x). Specifically, u_1'(x) = -sin(x) * sin(2x) / W(x) and u_2'(x) = cos(x) * sin(2x) / W(x), where W(x) = cos(x) * sin(x) - sin(x) * cos(x) = 1.
5. Step 5: Combine the homogeneous solution y_h(x) and the particular solution y_p(x) to form the general solution y(x). After performing the necessary integrations and simplifications, the general solution is y(x) = C_1 cos(x) + C_2 sin(x) - (1/3) sin(2x).