Question

The general solution of the differential equation: \[ (1 + \tan y) (dx - dy) + 2x \, dy = 0. \]

• A. \( e^x(y \cos x + \sin x) + \sin x = c \)
• B. \( e^x(y \cos x + y \sin x - \sin x) + \cos x = 0 \)
• C. \( e^y(x \cos y + x \sin y - \sin y) = c \)
• D. \( e^y(x \cos y + x \sin y + \sin y) = c \)

Hint:

For solving differential equations, first express in standard form \( \frac{dx}{dy} + P(x) = Q(y) \), then find the integrating factor and use it to solve the equation systematically.

Answer 1

The Correct Option is C

Step 1: Rearranging the given differential equation The given equation is: \[ (1 + \tan y) (dx - dy) + 2x \, dy = 0. \] Rewriting in standard form: \[ (1 + \tan y)dx - (1 + \tan y) dy + 2x dy = 0. \] \[ (1 + \tan y)dx + (-1 - \tan y + 2x) dy = 0. \] Rearranging: \[ \frac{dx}{dy} = \frac{1 + \tan y}{1 + \tan y - 2x}. \]
Step 2: Finding the integrating factor Rewriting the equation: \[ \frac{dx}{dy} - \frac{1 + \tan y}{1 + \tan y - 2x} = 0. \] We introduce the integrating factor \( e^y \), multiplying throughout: \[ e^y dx = e^y \left( x \cos y + x \sin y - \sin y \right) dy. \]
Step 3: Integrating both sides The equation is now separable: \[ \int d(e^y x) = \int e^y (x \cos y + x \sin y - \sin y) dy. \] Integrating both sides: \[ e^y (x \cos y + x \sin y - \sin y) = C. \] % Final Answer Thus, the correct answer is option (3): \( e^y (x \cos y + x \sin y - \sin y) = c \).