The general solution of $\frac{dy}{dx} + y \tan x = \sec x$ is

$y \sec x = \tan x + C$

The correct answer is Option (1) → $y \sec x = \tan x + C$ ##

Given differential equation is $\frac{dy}{dx} + y \tan x = \sec x$

which is a linear differential equation, comparing it with $\frac{dy}{dx} + Py = Q$, we get

$P = \tan x, Q = \sec x$  $[∵ e^{\log x}=x]$

$∴\text{I.F} = e^{\int \tan x \, dx} = e^{\log |\sec x|} = \sec x$

The general solution is $y \cdot \text{I.F} = \int (Q \cdot \text{I.F}) dx + C$

$y \cdot \sec x = \int \sec x \cdot \sec x \, dx + C$

$\Rightarrow y \sec x = \int \sec^2 x \, dx + C$

$\Rightarrow y \sec x = \tan x + C$