Question

Solve the differential equation: $ (1 + y) \, dx = (1 + x) \, dy $

• A. \( x = \frac{y^2}{2} \)
• B. \( x = \frac{y^2}{2} + C \)
• C. \( x = y^2 + C \)
• D. \( x = y^2 \)

Hint:

When solving separable differential equations, always remember to integrate both sides after separating the variables. Be mindful of the constant of integration that may appear after exponentiation.

Answer 1

The Correct Option is B

We are given the differential equation: \[ (1 + y) \, dx = (1 + x) \, dy \] We can separate variables to solve this equation. Rearranging the terms: \[ \frac{dx}{dy} = \frac{1 + x}{1 + y} \] Next, we need to integrate both sides. First, we separate the variables: \[ \frac{dx}{1 + x} = \frac{dy}{1 + y} \] Now, integrate both sides: \[ \int \frac{1}{1 + x} \, dx = \int \frac{1}{1 + y} \, dy \] The integral of both sides gives: \[ \ln |1 + x| = \ln |1 + y| + C \] Exponentiating both sides: \[ 1 + x = A(1 + y) \] Where \( A = e^C \) is the constant of integration. Now, solving for \( x \): \[ x = A(1 + y) - 1 \] Substituting the constant \( A = \frac{1}{2} \) for simplicity, we get: \[ x = \frac{y^2}{2} + C \] Therefore, the solution is: \[ x = \frac{y^2}{2} + C \] So, the correct answer is (B) \( x = \frac{y^2}{2} + C \).