Question

An integrating factor of the equation $(1+y+x^{2}y)dx+(x+x^3)dy=0$ is:

• A. $e^x$
• B. $x^2$
• C. $\frac{1}{x}$
• D. $x$

Answer 1

Answer: (D)

$x$

Dividing the entire equation by '$dx$' gives us
$(1+y+x^{2}y)+\frac{dy}{dx}(x+x^3)=0$
$\frac{dy}{dx}+\frac{y(1+x^2)}{x+x^3}=\frac{-1}{x+x^3}$
$\frac{dy}{dx}+\frac{y(1+x^2)}{x(1+x^2)}=\frac{-1}{x+x^3}$
$\frac{dy}{dx}+\frac{y}{x}=\frac{-1}{x+x^3}$
$\frac{dy}{dx}+yP\left(x\right)=Q\left(x\right)$
Hence, $IF=e^{\int P\left(x\right)dx}$
$=e^{\int \frac{1}{x}dx}$
$=e^{\log \left(x\right)}$
$=x$