Question

The general solution of the differential equation \((1 + \sin^2 x) \, \frac{dy}{dx} + \sin 2x = 0\) is?

• A. \((\sin 2x) y = \sin x + \sin^2 x + c\)
• B. \((\sin 2x) y = \sin x + \sin^2 x + c\)
• C. \((1 + \sin^2 x) y = \sin x + \sin^2 x + c\)
• D. \((\sin 2x) y = \sin 2x \)

Hint:

For solving these types of equations, remember to use standard integration techniques and simplify where possible.

Answer 1

The Correct Option is A

First, separate the terms and integrate both sides. The equation becomes: \[ (1 + \sin^2 x) \frac{dy}{dx} = -\sin 2x. \] Now, integrate both sides, solving for \(y\). This gives the solution: \[ (\sin 2x) y = \sin x + \sin^2 x + c. \]