Question

For the differential equation \(x^{\alpha} y^{\beta} (m y \, dx + n x \, dy) = 0\), the integrating factor, for any value of \(k\) is:

• A. \(x^{m - k - \alpha} y^{n - k - \beta}\)
• B. \(x^{m - 1 - k\alpha} y^{n - 1 - k\beta}\)
• C. \(x^{km - 1 - \alpha} y^{kn - 1 - \beta}\)
• D. \(x^{km - \alpha} y^{kn - \beta}\)

Hint:

Exam Tip:
For equations of the form \(x^{\alpha} y^{\beta} (m y \, dx + n x \, dy) = 0\):

• The integrating factor is \(x^{km - 1 - \alpha} y^{kn - 1 - \beta}\).
• This is a standard result for such equations.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to find the integrating factor for a differential equation of a specific form. The equation is \(x^{\alpha} y^{\beta} (m y \, dx + n x \, dy) = 0\).

Step 2: Key Formula or Approach:
The equation can be written as: \[ m x^{\alpha} y^{\beta + 1} \, dx + n x^{\alpha + 1} y^{\beta} \, dy = 0 \] This is of the form \(M \, dx + N \, dy = 0\), where: \[ M = m x^{\alpha} y^{\beta + 1}, \quad N = n x^{\alpha + 1} y^{\beta} \] We need to find an integrating factor \(I = x^p y^q\) that makes the equation exact.

Step 3: Detailed Explanation:
For the equation to be exact after multiplying by \(I\): \[ \frac{\partial}{\partial y} (I M) = \frac{\partial}{\partial x} (I N) \] Let \(I = x^p y^q\). Then: \[ \frac{\partial}{\partial y} (x^p y^q \cdot m x^{\alpha} y^{\beta + 1}) = \frac{\partial}{\partial x} (x^p y^q \cdot n x^{\alpha + 1} y^{\beta}) \] \[ \frac{\partial}{\partial y} (m x^{p + \alpha} y^{q + \beta + 1}) = \frac{\partial}{\partial x} (n x^{p + \alpha + 1} y^{q + \beta}) \] \[ m x^{p + \alpha} (q + \beta + 1) y^{q + \beta} = n (p + \alpha + 1) x^{p + \alpha} y^{q + \beta} \] Cancel \(x^{p + \alpha} y^{q + \beta}\): \[ m(q + \beta + 1) = n(p + \alpha + 1) \] We need to find \(p\) and \(q\) in terms of \(k\) (a parameter).
This equation gives one relation between \(p\) and \(q\).
The integrating factor is of the form \(x^p y^q\).
The options suggest that \(p = km - 1 - \alpha\) and \(q = kn - 1 - \beta\).
Let's check if these satisfy the condition: \[ m(q + \beta + 1) = m(kn - 1 - \beta + \beta + 1) = m(kn) = k m n \] \[ n(p + \alpha + 1) = n(km - 1 - \alpha + \alpha + 1) = n(km) = k m n \] So, the condition holds for any \(k\).
Thus, the integrating factor is \(x^{km - 1 - \alpha} y^{kn - 1 - \beta}\).
This matches option (C).

Step 4: Final Answer:
Therefore, option (C) is correct.