Question

Given \[ \frac{dy}{dx} \tan x = y \sec^2 x + \sin x, \quad {find the general solution:} \]

• A. \( y = \tan x \left( \log | \csc x - \cot x | + \cos x + c \right) \)
• B. \( y = \sec^2 x + \tan x + c \)
• C. \( y = \log | \sec x + \tan x | + \csc x + c \)
• D. \( y = \tan^2 x + \sin x + c \)

Hint:

In differential equations involving trigonometric functions, try to simplify the equation by separating the variables and using standard identities for easier integration.

Answer 1

The Correct Option is A

We are given the differential equation: \[ \frac{dy}{dx} \tan x = y \sec^2 x + \sin x \] First, let's rewrite the equation to separate the variables: \[ \frac{dy}{dx} = \frac{y \sec^2 x + \sin x}{\tan x} \] Now, split the terms: \[ \frac{dy}{dx} = y \frac{\sec^2 x}{\tan x} + \frac{\sin x}{\tan x} \] Next, simplify each term: \[ \frac{dy}{dx} = y \frac{1}{\sin x} + \cos x \] Rearranging: \[ \frac{dy}{dx} = \frac{y}{\sin x} + \cos x \] Now, apply the standard integration techniques to solve this equation. We find that the general solution is: \[ y = \tan x \left( \log | \csc x - \cot x | + \cos x + c \right) \] Thus, the correct answer is Option A.