Question

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

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

Hint:

When solving differential equations, always simplify and check for separable variables.

Answer 1

The Correct Option is A

Step 1: Simplify the differential equation.
The given equation can be simplified by separating the variables and integrating. After solving, we find the solution as \( (x - 2) = k e^{-\tan^{-1}y} \).
Step 2: Conclusion.
The correct answer is (A).