Question

The integrating factor of the linear DE \(\dfrac{dy}{dx}+\dfrac{2}{x}y=5x^{2}\) is

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

Hint:

For \(y'+\dfrac{m}{x}y=\ldots\), IF is \(x^{m}\).

Answer 1

The Correct Option is D

Standard form \(y'+P(x)y=Q(x)\) has IF \(=\exp\!\left(\int P(x)\,dx\right)\). Here \(P(x)=\dfrac{2}{x}\). So \[ \text{IF}=\exp\!\left(\int \frac{2}{x}dx\right)=\exp(2\ln x)=e^{\ln x^{2}}=x^{2}. \]