Consider the differential equation $\frac{dy}{dx} + y \tan x = \sec x$, then which of the following statements are correct?

(A) It is homogeneous

(B) It has $\sec x$ as its integrating factor

(C) It's general solution is $y \sec x = \tan x + c$, where c is arbitary constant.

(D) It's degree is not defined

Choose the correct answer from the options given below:

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

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Statement (A): It is homogeneous

A differential equation is homogeneous if it can be written as $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$.

The equation becomes: $\frac{dy}{dx} = \sec x - y \tan x$

This contains $\sec x$ and $\tan x$ (functions of $x$ only), not in the form $f\left(\frac{y}{x}\right)$.

Statement A is FALSE.

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Statement (B): It has $\sec x$ as its integrating factor

For a linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$, the integrating factor is:

$IF = e^{\int P(x)dx}$

Here, $P(x) = \tan x$

$IF = e^{\int \tan x \, dx}$

$\int \tan x \, dx = \int \frac{\sin x}{\cos x} dx$

$= -\ln|\cos x|$

$= \ln|\sec x|$

Therefore:

$IF = e^{\ln|\sec x|}$

$= \sec x$

Statement B is TRUE.

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Statement (C): General solution is $y \sec x = \tan x + c$

Multiply the entire equation by $\sec x$:

$\sec x \cdot \frac{dy}{dx} + y \sec x \tan x = \sec^2 x$

The left side is $\frac{d}{dx}(y \sec x)$ since:

$\frac{d}{dx}(y \sec x) = \frac{dy}{dx} \sec x + y \cdot \sec x \tan x$

Therefore:

$\frac{d}{dx}(y \sec x) = \sec^2 x$

Integrating both sides:

$y \sec x = \int \sec^2 x \, dx$

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

Statement C is TRUE.

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Statement (D): Its degree is not defined

The degree is the power of the highest order derivative when the equation is expressed as a polynomial in derivatives.

In $\frac{dy}{dx} + y \tan x = \sec x$, the derivative $\frac{dy}{dx}$ appears to the power 1.

The degree is 1 (it is defined).

Statement D is FALSE.

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Statements (B) and (C) only are correct.