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  4. The integrating factor of (dy/dx) + y = (1 + y/x) is

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

KCETKCET 2018Differential Equations

Solution:

Given $\frac{dy}{dx} +y = \frac{1+y}{x} $
$\Rightarrow \frac{dy}{dx} +y =\frac{1}{x} +\frac{y}{x} $
$\Rightarrow \frac{dy}{dx} + \left(1 - \frac{1}{x}\right)y = \frac{1}{x} $
$ \therefore IF -e^{\int\left(1 - \frac{1}{x}\right)dx} = e^{x -\log x} = \frac{e^{x}}{x}$