Question

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

• A. \(x^3\)
• B. \(x^2\)
• C. \(log2x\)
• D. \(logx^2\)

Hint:

Divide by x to get standard form dy/dx + (2/x)y = x log x, then IF = e^(integral of 2/x).

Answer 1

The Correct Option is B

Step 1: Standard form first:
Divide by $x$: $y' + \frac{2}{x}y = x\ln x$. Here $P(x) = \frac2x$.

Step 2: Multiply by a trial factor:
Take $\mu = x^2$. Then $x^2y' + 2xy = (x^2y)'$, which is a perfect derivative. So $\mu = x^2$ makes the left side exact, which is the job of an integrating factor.

Step 3: Confirm with the formula:
$\mu = \exp\left(\int\frac2x\,dx\right) = e^{2\ln x} = x^2$.

Final Answer:
$x^2$, option (B). \[ \boxed{x^2 \text{ (B)}} \]