Question

The integrating factor of the differential equation \[ \frac{dy}{dx} + \frac{1}{x}y = x^3 - 3 \] is

• A. \( -y \)
• B. \( y \)
• C. \( x \)
• D. \( -x \)

Hint:

For linear differential equations, the integrating factor depends only on the coefficient of \( y \).

Answer 1

The Correct Option is C

Step 1: Identify the standard linear form.
The given equation is of the form \[ \frac{dy}{dx} + P(x)y = Q(x) \] where \[ P(x) = \frac{1}{x} \]

Step 2: Formula for integrating factor.
\[ \text{I.F.} = e^{\int P(x)\,dx} \]

Step 3: Compute the integrating factor.
\[ \text{I.F.} = e^{\int \frac{1}{x}dx} = e^{\ln x} = x \]

Step 4: Conclusion.
The integrating factor is \[ \boxed{x} \]