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Question
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is
Options
sin x
sec x
tan x
cos x
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Solution
sec x
We have,
\[\cos x\frac{dy}{dx} + y \sin x = 1\]
Dividing both sides by cos x, we get
\[\frac{dy}{dx} + \frac{\sin x}{\cos x}y = \frac{1}{\cos x}\]
\[ \Rightarrow \frac{dy}{dx} + \left( \tan x \right)y = \frac{1}{\cos x}\]
\[\text{Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get }\]
\[P = \tan x\]
\[Q = \frac{1}{\cos x}\]
Now,
\[ I . F . = e^{\int\tan xdx} = e^{log\left( \sec x \right)} \]
\[ = \sec x\]
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