The integrating factor of the differential equation x dy dx -y=2x…

MCQ

The integrating factor of the differential equation $\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=2\text{x}^{2}.$

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

✓

Answer

Correct option: C.

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

We have,
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=2\text{x}^{2}$
$\Rightarrow \frac{\text{dy}}{\text{dx}}-\frac{1}{\text{x}}\text{y}=2\text{x}$
Comparing with $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$ we get,
$\text{P}=-\frac{1}{\text{x}}, \text{Q}=2\text{x}$
Now,
$\text{I.F}=\text{e}^{\int\frac{1}{\text{x}}\text{dy}}$
$=\text{e}^{\log|\text{x}|}$
$=\text{e}^{\log|\frac{1}{\text{x}}|}$
$=\frac{1}{\text{x}}$

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