Question

Integrating factor of the linear differential equation $\frac{dy}{dx} + \frac{2y}{x} = x \log x$ is

• A. $x$
• B. $x^2$
• C. $x^3$
• D. $x^4$

Hint:

The integrating factor for \( \frac{dy}{dx} + P(x)y = Q(x) \) is \( e^{\int P(x) \, dx} \); compute the integral carefully to determine the correct power of \( x \).

Answer 1

The Correct Option is B

Step 1: Identify the form of the differential equation.
The given equation is: \[ \frac{dy}{dx} + \frac{2y}{x} = x \log x. \] This is a first-order linear differential equation of the form: \[ \frac{dy}{dx} + P(x) y = Q(x), \] where:
\( P(x) = \frac{2}{x} \),
\( Q(x) = x \log x \).
Step 2: Find the integrating factor.
The integrating factor (IF) for a first-order linear differential equation is: \[ \text{IF} = e^{\int P(x) \, dx}. \] Compute the integral of \( P(x) \): \[ P(x) = \frac{2}{x}, \quad \int P(x) \, dx = \int \frac{2}{x} \, dx = 2 \log x = \log x^2, \] \[ \text{IF} = e^{\int P(x) \, dx} = e^{\log x^2} = x^2. \] Step 3: Evaluate the options.
(1) \( x \): Incorrect, as the integrating factor is \( x^2 \), not \( x \). Incorrect.
(2) \( x^2 \): Correct, as the integrating factor is \( x^2 \). Correct.
(3) \( x^3 \): Incorrect, as the integrating factor is \( x^2 \), not \( x^3 \). Incorrect.
(4) \( x^4 \): Incorrect, as the integrating factor is \( x^2 \), not \( x^4 \). Incorrect.
Step 4: Select the correct answer.
The integrating factor is \( x^2 \), matching option (2).