Question

If \(g(x)\) denotes the integrating factor of the differential equation \(x\log x\frac{dy}{dx}+y=2\,\log x\) and \(\int g(x)dx=h(x)[g(x)-1]+c\), then \(h(x)=\).........

• A. \(1\)
• B. \(x\)
• C. \(\log x\)
• D. \(x^{2}\)

Hint:

Convert to linear form to find the integrating factor, then integrate it and compare with the given relation.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Divide by \(x\log x\) to get the linear form \(\dfrac{dy}{dx}+\dfrac{1}{x\log x}y=\dfrac2x\).

Step 2: Integrating factor
\(g(x)=e^{\int\frac{dx}{x\log x}}=e^{\log(\log x)}=\log x\).

Step 3: Integrate g(x)
\(\int\log x\,dx=x\log x-x+c=x(\log x-1)+c\).

Step 4: Compare
Since \(g(x)=\log x\), this is \(x\,[g(x)-1]+c\), so \(h(x)=x\).

Final Answer:
\(h(x)=x\). \[ \boxed{\text{(B) }x} \]