Question

The general solution of the differential equation \( \sec(x - y + 1) dy = dx \) is:

• A. \( x + \cot\left(\frac{x - y + 1}{2}\right) = c \)
• B. \( x + \cot(x - y + 1) = c \)
• C. \( x - \cot\left(\frac{x - y + 1}{2}\right) = c \)
• D. \( x - \cot(x - y + 1) = c \)

Hint:

For separable differential equations, rewrite them in the form \( g(y) dy = f(x) dx \), integrate both sides separately, and simplify.

Answer 1

The Correct Option is B

Rewriting: \[ \sec(x - y + 1) \, dy = dx \] Separating variables: \[ dy = \frac{dx}{\sec(x - y + 1)} \] \[ dy = \cos(x - y + 1) \, dx \] Integrating both sides: \[ \int dy = \int \cos(x - y + 1) \, dx \] Using standard trigonometric integration: \[ y = x + \cot(x - y + 1) + c \] Thus, the correct answer is: \[ x + \cot(x - y + 1) = c \]