Question

The integrating factor of the differential equation $(3x^2 + y) \frac{dx}{dy} = x$ is:

• A. $\frac{1}{x}$
• B. $\frac{1}{x^2}$
• C. $\frac{2}{x}$
• D. $-\frac{1}{x}$

Hint:

Write in standard linear or separable form to find IF.

Answer 1

The Correct Option is A

Rearrange: \[ \frac{dx}{dy} - \frac{1}{3x^2+y} \cdot x = 0. \text{But easier to separate: treat as linear in } x. \] The linear form: \[ \frac{dx}{dy} - \frac{1}{3x^2+y} x = 0. \] Wait! There is a simpler approach if written properly: \[ \frac{dx}{dy} = \frac{x}{3x^2 + y}. \] This requires an integrating factor. It is $\frac{1}{x}$ (direct separation works).