Choose the correct answer from the given four option. Integrating… - Vidyadip

Question

Choose the correct answer from the given four option.
Integrating factor of the differential equation $\frac{\text{d}\text{y}}{\text{d}\text{x}}+\text{y}\tan\text{x}-\sec\text{x}=0$ is:

$\cos\text{x}$
$\sec\text{x}$
$\text{e}^{\cos\text{x}}$
$\text{e}^{\sec\text{x}}$

✓

Answer

$\sec\text{x}$

Solution:
Given that, $\frac{\text{d}\text{y}}{\text{d}\text{x}}+\text{y}\tan\text{x}-\sec\text{x}=0$
$\Rightarrow\frac{\text{d}\text{y}}{\text{d}\text{x}}+\text{y}\tan\text{x}=\sec\text{x}$
$\Big($It is a linear differential equation of form $\frac{\text{d}\text{y}}{\text{d}\text{x}}+\text{P}\text{y}=\text{Q}\Big)$
Here, $\text{P}=\tan\text{x},\text{Q}=\sec\text{x}$
$=\text{I.F.}=\text{e}^{\int\text{Pdx}}$
$=\text{e}^{\int\tan\text{x}\text{dx}}$
$=\text{e}^{(\log\sec\text{x})}$
$=\sec\text{x}$

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