Question

The integrating factor of the differential equation $\frac{dy}{dx}=\frac{1}{x+y+2}$ is

• A. $e^{x+y+2}$
• B. $e^y$
• C. $e^{-y}$
• D. $\log |x+y+2|$

Answer 1

Answer: (C)

$e^{-y}$

Given D.E is $\frac{dy}{dx}=\frac{1}{x+y+2}$

$\Rightarrow \frac{dx}{dy}=x+y+2$
$\Rightarrow \frac{dx}{dy}-x=y+2$ ....... $\left(i\right)$

which is a linear D.E of the form $\frac{dx}{dy}+P\left(y\right)x=Q\left(y\right)$

From $\left(i\right)$, $P\left(y\right)=-1,Q\left(y\right)=y+2$
$\therefore I.F=e^{\int P\left(y\right)dy}=e^{\int -dy}=e^{-y}$.