Question

Find the general solution of the differential equation \( ( \sin y \cos^2 y - x \sec^2 y ) dy = (\tan y) dx \).

• A. \( \tan y = 3x \cos^3 y + c \)
• B. \( x (\sec y + \tan y) = \cos^2 y + c \)
• C. \( y \sin y = x^2 \cos^2 y + c \)
• D. \( 3x \tan y + \cos^3 y = c \)

Hint:

For solving first-order differential equations, separate the variables properly and integrate both sides step-by-step.

Answer 1

The Correct Option is D

Step 1: Given differential equation. We start with the given equation: \[ ( \sin y \cos^2 y - x \sec^2 y ) dy = (\tan y) dx \] Step 2: Separating the variables. Rewriting the equation: \[ \frac{dy}{dx} = \frac{\tan y}{\sin y \cos^2 y - x \sec^2 y} \] Rearranging terms to make it integrable: \[ \int (\sin y \cos^2 y - x \sec^2 y) dy = \int \tan y \, dx \] Step 3: Integrating both sides. Integrating LHS: \[ \int \sin y \cos^2 y \, dy - \int x \sec^2 y \, dy \] The first integral simplifies to: \[ \frac{\cos^3 y}{3} \] The second integral simplifies to: \[ x \tan y \] Thus, we get: \[ 3x \tan y + \cos^3 y = c \]