Question

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

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

Hint:

To solve separable differential equations, separate variables and integrate each side independently.

Answer 1

The Correct Option is B

Step 1: Solving the differential equation.
The given differential equation is separable and can be solved by separation of variables. After solving, we find that the solution is \( x \cdot e^{2y} = e^y + K \).

Step 2: Conclusion.
The correct solution is \( x \cdot e^{2y} = e^y + K \), corresponding to option (2).