Question

An integrating factor of the differential equation
\[ \left( y + \frac{1}{3} y^3 + \frac{1}{2} x^2 \right) \, dx + \frac{1}{4} (x + x^2)^2 \, dy = 0 \] is

• A. \( x^2 \)
• B. \( 3 \log x \)
• C. \( x^3 \)
• D. \( 2 \log x \)

Hint:

For non-exact differential equations, the integrating factor often depends on the variable that causes the equation to become exact.

Answer 1

The Correct Option is C

Step 1: Identifying the type of equation.
We recognize that the given differential equation is linear, and we need to find an integrating factor. The integrating factor is usually a function of \( x \) or \( y \) that makes the equation exact.

Step 2: Finding the integrating factor.
The integrating factor for this equation is found to be \( x^2 \), which simplifies the equation to an exact differential.

Step 3: Conclusion.
The correct answer is (A) \( x^2 \).