Question

The integrating factor of the differential equation $x.\frac{dy}{dx}+2y=x^2$ is $(x\ne 0)$:

• A. x
• B. log |x|
• C. $e^{log.x}$
• D. $x^2$

Answer 1

The correct option is D $x^2$

$x\cdot \frac{dy}{dx}+2y=x^2$

$\frac{dy}{dx}+\frac{2}{x}\cdot y=x$
Comparing above equation with $\frac{dy}{dx}+Py=Q$

We get, $P=\frac{2}{x},Q=x$
Integrating factor, $I=e^{\int Pdx}$

$I=e^{\int \frac{2}{x}dx}$
Now, $\int \frac{2}{x}dx=2{\log }_{e}x$

$\therefore I=e^{2{\log }_{e}x}$

$I=e^{{\log }_e{x}^2}$

$I=x^2$