Question

Integrating factor of the differential equation cos $x\frac{dy}{dx}+y\sin x=1$, is
(a) cos x
(b) tan x
(c) sec x
(d) sin x

Answer 1

(c) sec x

We have,
$\cos x\frac{dy}{dx}+y\sin x=1$
Dividing both sides by cos x, we get
$$\begin{gathered} \frac{dy}{dx}+\frac{\sin x}{\cos x}y=\frac{1}{\cos x} \\ \Rightarrow \frac{dy}{dx}+\left(\tan x\right)y=\frac{1}{\cos x} \\ \mathrm{Comparing}\mathrm{with}\frac{dy}{dx}+Py=Q,\mathrm{we}\mathrm{get} \\ P=\tan x \\ Q=\frac{2}{\cos x} \\ \mathrm{Now}, \\ \mathrm{I}.\mathrm{F}.=e^{\int \tan xdx} \\ =e^{\log \left(se\mathrm{c}x\right)} \\ =\sec x \end{gathered}$$