Consider the differential equation \(\displaystyle\frac{dy}{dx}+y\tan x=\sec x\), then which of the following statements are correct?
(A) It is homogeneous
(B) It has \(\sec x\) as its integrating factor
(C) Its general solution is \(y \sec x = \tan x + c\), where c is arbitrary constant.
(D) Its degree is not defined
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (C) only
The equation is linear in y, of the form \(\frac{dy}{dx} + Py = Q\) with P = tan x and Q = sec x.
(A) It is homogeneous — Incorrect. Here \(\frac{dy}{dx} = \sec x - y\tan x\), which cannot be written as a function of \(\frac{y}{x}\) alone.
(B) It has \(\sec x\) as its integrating factor — Correct. With P = tan x:
\(\displaystyle \text{I.F.} = e^{\int \tan x\,dx} = e^{\log|\sec x|} = \sec x\)
(C) Its general solution is \(y \sec x = \tan x + c\), where c is arbitrary constant. — Correct. Multiplying by the I.F. and integrating:
\(\displaystyle y \sec x = \int \sec x \cdot \sec x\,dx = \int \sec^2 x\,dx\)
\(y \sec x = \tan x + c\)
(D) Its degree is not defined — Incorrect. The equation is a polynomial in \(\frac{dy}{dx}\), whose highest power is 1, so its degree is 1. (The tan x and sec x are in x, not in the derivative.)