Question

For the differential equation
y(8x − 9y)dx + 2x(x − 3y)dy = 0 ,
which one of the following statements is TRUE ?

• A. The differential equation is not exact and has x² as an integrating factor
• B. The differential equation is exact and homogeneous
• C. The differential equation is not exact and does not have x² as an integrating factor
• D. The differential equation is not homogeneous and has x² as an integrating factor

Answer 1

The Correct Option is A

The given differential equation is: \[ M(x, y) \, dx + N(x, y) \, dy = 0, \] where \[ M(x, y) = y(8x - 9y), \quad N(x, y) = 2x(x - 3y). \]

For exactness, we need to check if the condition \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \) holds: \[ \frac{\partial M}{\partial y} = 8x - 18y, \quad \frac{\partial N}{\partial x} = 2(x - 3y) + 2x = 4x - 6y. \] Since \( \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x} \), the differential equation is not exact.

Next, we check if \( x^2 \) is an integrating factor. Multiply both \( M(x, y) \) and \( N(x, y) \) by \( x^2 \): \[ M'(x, y) = x^2 y(8x - 9y) = 8x^3 y - 9x^2 y, \quad N'(x, y) = x^2 \cdot 2x(x - 3y) = 2x^3(x - 3y). \]

Now check exactness for the modified equation: \[ \frac{\partial M'}{\partial y} = 8x^3 - 9x^2, \quad \frac{\partial N'}{\partial x} = 3x^2(x - 3y) + 2x^3 = 8x^3 - 9x^2. \] Since \( \frac{\partial M'}{\partial y} = \frac{\partial N'}{\partial x} \), the equation becomes exact with the integrating factor \( x^2 \).

Thus, the correct answer is (A): The differential equation is not exact and has \( x^2 \) as an integrating factor.