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

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

Answer: (A)

Verified step by step guidance

1. Step 1: Identify the type of differential equation. The given equation is a second-order linear differential equation: y'' + y = sec(x) tan(x). The right-hand side sec(x) tan(x) is a non-homogeneous term.
2. Step 2: Solve the corresponding homogeneous equation y'' + y = 0. The characteristic equation for this is r^2 + 1 = 0, which has roots r = ±i. This implies the general solution to the homogeneous equation is y_h = C_1 cos(x) + C_2 sin(x), where C_1 and C_2 are constants.
3. Step 3: Apply the method of variation of parameters to find a particular solution y_p for the non-homogeneous equation. Variation of parameters involves using the solutions of the homogeneous equation (cos(x) and sin(x)) to construct a particular solution. Assume y_p = 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: Derive expressions for u_1(x) and u_2(x) using the formulas: u_1'(x) = -[sin(x) * sec(x) tan(x)] / W and u_2'(x) = [cos(x) * sec(x) tan(x)] / W, where W is the Wronskian of cos(x) and sin(x), given by W = cos(x) * sin'(x) - sin(x) * cos'(x) = 1.
5. Step 5: Integrate u_1'(x) and u_2'(x) to find u_1(x) and u_2(x). Substitute these into y_p = u_1(x) cos(x) + u_2(x) sin(x) to obtain the particular solution. Combine y_h and y_p to form the general solution: y = C_1 cos(x) + C_2 sin(x) + (particular solution).