Question

Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is

Options

• cos x

• tan x

• sec x

• sin x

Answer 1

sec x

 

We have,

\[\cos x\frac{dy}{dx} + y \sin x = 1\]

Dividing both sides by cos x, we get

\[\frac{dy}{dx} + \frac{\sin x}{\cos x}y = \frac{1}{\cos x}\]

\[ \Rightarrow \frac{dy}{dx} + \left( \tan x \right)y = \frac{1}{\cos x}\]

\[\text{ Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get }\]

\[P = \tan x\]

\[Q = \frac{2}{\cos x}\]

Now,

\[I . F . = e^{\int\tan xdx} \]

\[ = e^{log\left( sec x \right)} \]

\[ = \sec x\]