Consider the differential equation \(xdy = (x + y)\, dx\). Which of the following are true?
(A) It is a homogeneous differential equation
(B) It is a differential equation of order 2
(C) The general solution of the differential equation contains 2 arbitrary constants
(D) Integrating factor of differential equation is \(\displaystyle\frac{1}{x}\)
(E) Degree of the differential equation is not defined
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (D) only
Dividing by \(x\,dx\):
\( \displaystyle \frac{dy}{dx} = \frac{x + y}{x} = 1 + \frac{y}{x} \)
So \(\frac{dy}{dx} = 1 + \frac{y}{x}\): an equation of order 1 and degree 1, which is both homogeneous and linear.
Checking the statements
(A) It is a homogeneous differential equation — Correct. \(1 + \frac{y}{x}\) is a homogeneous function of degree 0.
(B) It is a differential equation of order 2 — Incorrect. The order is 1.
(C) The general solution of the differential equation contains 2 arbitrary constants — Incorrect. The order is 1, so there is 1 arbitrary constant.
(D) Integrating factor of differential equation is \(\displaystyle\frac{1}{x}\) — Correct. As a linear equation, \(\frac{dy}{dx} - \frac{1}{x}y = 1\), so \(P = -\frac{1}{x}\):
\( \displaystyle \text{I.F.} = e^{\int -\frac{1}{x}\,dx} = e^{-\log x} = \frac{1}{x} \)
(E) Degree of the differential equation is not defined — Incorrect. The degree is 1.