The integrating factor of the differential equation (x ) dy dx +y… - Vidyadip

MCQ

The integrating factor of the differential equation $(\text{x}\log\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}=2\ \log\text{x}$ is given by :

A
$\log(\log\text{x})$
B
$\text{e}^{\text{x}}$
✓
$\log\text{x}$
D
$\text{x}$

✓

Answer

Correct option: C.

$\log\text{x}$

We have,
$(\text{x}\log\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}=2\ \log\text{x}$
Dividing both sides by,
$\frac{\text{dy}}{\text{dx}}+\frac{\text{y}}{\text{x}+\log\text{x}}=\frac{2}{\text{x}}$
$\frac{\text{dy}}{\text{dx}}+\Big(\frac{\text{1}}{\text{x}+\log\text{x}}\Big)\text{y}=\frac{2}{\text{x}}$
Comparing with $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
$\text{P}=\frac{1}{\text{x}\log\text{x}}$
$\text{Q}=\frac{2}{\text{x}}$
Now,
$\text{I.F}=\text{e}^{\int\text{P}\text{dx}}$
$\text{I.F}=\text{e}^{\int\frac{1}{\text{x}\log\text{x}}}\text{dx}$
$=\text{e}^{\log(\log\text{x})}$
$=\log\text{x}$

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