The Integrating Factor of the differential equation x dy dx -y=2x…

MCQ

The Integrating Factor of the differential equation $\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=2\text{x}^2$ is

A
$\text{e}^{-\text{x}}$
B
$\text{e}^{-\text{y}}$
✓
$\frac{1}{\text{x}}$
D
$\text{x}$

✓

Answer

Correct option: C.

$\frac{1}{\text{x}}$

The given differential equation is :
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=2\text{x}^2$
$\Rightarrow\frac{\text{dy}}{\text{dx}}-\frac{\text{y}}{\text{x}}=2\text{x}$
This is a linear differential equation of the form :
$\frac{\text{dy}}{\text{dx}}+\text{py}=\text{Q}\ ($where ${p}=-\frac{1}{\text{x}}$ and $\text{Q}=2\text{x})$
The integrating factor $\text{(I.F)}$ is given by the relation,
$\text{e}^{\int\text{pdx}}$
$\therefore\text{I.F}=\text{e}^{\int-\frac{1}{\text{x}}\text{dx}}=\text{e}^{-\log\text{x}}=\text{e}^{\log(\text{x}^-1)}=\text{x}^{-1}=\frac{1}{\text{x}}$

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