Question

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

• A. \( \log x \)
• B. \( e^x \)
• C. \( \frac{1}{x} \)
• D. \( x \)

Hint:

The integrating factor for a first-order linear differential equation \( y' + P(x)y = Q(x) \) is always \( e^{\int P(x) dx} \). The key is to correctly identify the \( P(x) \) term, which is the coefficient of the \( y \) term.

Answer 1

The Correct Option is D

Step 1: Identify the form of the differential equation. The equation is in the linear form \( \frac{dy}{dx} + P(x)y = Q(x) \).
Step 2: Identify the function \( P(x) \). By comparing the given equation to the standard form, we find that \( P(x) = \frac{1}{x} \).
Step 3: Calculate the integrating factor (I.F.) using the formula I.F. = \( e^{\int P(x) dx} \). \[ \text{I.F.} = e^{\int \frac{1}{x} dx} \] Step 4: Evaluate the integral and simplify. \[ \text{I.F.} = e^{\ln x} = x \]