Question

Write the integrating factor of the differential equation:
$\cos x\frac{dy}{dx}+y=\sin x;0\le x<\frac{\pi }{2}$.

Answer 1

Suppose we have the first order differential equation

$\frac{dy}{dx}+Py=Q$

where$P$ and $Q$ are functions involving $x$ only.

We multiply both sides of the differential equation by the integrating factor$I$ which is defined as

$e^{\int Pdx}$

$\cos x\frac{dy}{dx}+y=sinx$

$⟹\frac{dy}{dx}+y\sec x=\tan x$

Here $P=\sec x$

Hence, the integrating factor $I$ becomes $e^{\int \sec xdx}=e^{\ln |\sec x+\tan x|}=|\sec x+\tan x|=\sec x+\tan x$, because both $secx,tanx$ are positive in first quadrant.