Question

The integrating factor of $ x\frac{dy}{dx}+(1+x)y=x $ is

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

Answer 1

Answer: (E)

Given differential equation is $ x\frac{dy}{dx}+(1+x)y=x $
$ \Rightarrow $ $ \frac{dy}{dx}+\left( \frac{1+x}{x} \right)y=1 $
Which is linear differential equation Hence, $ IF={{e}^{\int{\frac{1+x}{x}dx}}}={{e}^{\int{\left( \frac{1}{x}+1 \right)}dx}} $
$={{e}^{\log x+x}} $
$={{e}^{\log x}}.{{e}^{x}} $
$=x{{e}^{x}} $