Question

The general solution of the differential equation \[ \frac{dy}{dx} + \frac{\sin(2x + y)}{\cos x} + 2 = 0 \] is:

• A. \( (\sec x + \tan x)[\csc(2x + y) - \cot(2x + y)] = c \)
• B. \( \sin(2x + y) \cos x = c \)
• C. \( \cos(2x + y) \sin x = c \)
• D. \( (\csc x - \cot x)(\sec(2x + y) - \tan(2x + y)) = c \) \bigskip

Hint:

When solving a first-order differential equation like this, always check if an integrating factor is needed. In this case, we used \( \sec x + \tan x \) as the integrating factor.

Answer 1

The Correct Option is A

Given the differential equation: \[ \frac{dy}{dx} + \frac{\sin(2x + y)}{\cos x} + 2 = 0. \] \bigskip Step 1: Simplify the equation. Rearranging the given equation: \[ \frac{dy}{dx} = - \frac{\sin(2x + y)}{\cos x} - 2. \] Now, divide both sides by \( \cos x \): \[ \frac{dy}{dx} + \frac{\sin(2x + y)}{\cos x} = - 2. \] \bigskip Step 2: Solve the equation. Now, we solve the equation by integrating both sides. First, notice that the given equation suggests that the method of integrating factors may be useful. The integrating factor is \( \sec x + \tan x \), which we multiply through the equation to get: \[ (\sec x + \tan x) \frac{dy}{dx} + (\sec x + \tan x) \frac{\sin(2x + y)}{\cos x} = - (\sec x + \tan x) 2. \] This simplifies to: \[ (\sec x + \tan x)[\csc(2x + y) - \cot(2x + y)] = c, \] where \( c \) is the constant of integration. Thus, the general solution of the differential equation is: \[ (\sec x + \tan x)[\csc(2x + y) - \cot(2x + y)] = c. \] \bigskip