Question

If \( x^h y^k \) is an integrating factor of the differential equation \[ y(1 + xy) \, dx + x(1 - xy) \, dy = 0, \] then the ordered pair \( (h, k) \) is equal to

• A. \( (-2, -2) \)
• B. \( (-2, -1) \)
• C. \( (-1, -2) \)
• D. \( (-1, -1) \)

Hint:

When solving for integrating factors, make sure to check the exactness condition by calculating the mixed partial derivatives and comparing them.

Answer 1

The Correct Option is A

Step 1: Analyze the differential equation.
The given differential equation is: \[ y(1 + xy) \, dx + x(1 - xy) \, dy = 0. \] We are tasked with finding an integrating factor of the form \( x^h y^k \). To find the integrating factor, we multiply the whole equation by \( x^h y^k \). The goal is to make the resulting equation exact, meaning the total derivative with respect to some potential function must be zero.
Step 2: Compute the derivatives.
For the given equation to become exact, the following conditions must be satisfied: \[ \frac{\partial}{\partial y} \left( x^h y^k (y(1 + xy)) \right) = \frac{\partial}{\partial x} \left( x^h y^k (x(1 - xy)) \right). \] From this, solve for \( h \) and \( k \) by simplifying the expressions. By performing this analysis, we find that \( h = -2 \) and \( k = -1 \).

Step 3: Conclusion.
Thus, the ordered pair \( (h, k) \) is \( (-2, -1) \), so the correct answer is \( (B) \).