Question

The solution of the differential equation \[ (1 + y^2) + (x - e^{\tan^{-1} y}) \frac{dy}{dx} = 0 \] is:

• A. \( 2x e^{\tan^{-1} y} = e^{2\tan^{-1} y} + C \)
• B. \( x e^{\tan^{-1} y} = \tan^{-1} y + C \)
• C. \( x e^{\tan^{-1} y} = e^{\tan^{-1} y} + C \)
• D. \( (x - 2) = C e^{-\tan^{-1} y} \)

Hint:

When solving first-order differential equations, always try to isolate the terms involving \( \frac{dy}{dx} \) and integrate both sides.

Answer 1

The Correct Option is A

We are given the differential equation: \[ (1 + y^2) + (x - e^{\tan^{-1} y}) \frac{dy}{dx} = 0 \] Step 1: Solve the equation Rearranging the terms and solving the equation, we get the solution: \[ 2x e^{\tan^{-1} y} = e^{2\tan^{-1} y} + C \] Thus, the correct answer is \( 2x e^{\tan^{-1} y} = e^{2\tan^{-1} y} + C \).