Consider the differential equation $xdy = (x + y) dx$. Which of the following are true?

(A) It is a homogenous 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 $\frac{1}{x}$

(E) Degree of the differential equation is not defined

Choose the correct answer from the options given below:

Given: $xdy = (x + y)dx$

Dividing both sides by $dx$:

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

Dividing by $x$:

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

$\frac{dy}{dx} = 1 + \frac{y}{x}$

This can also be written in linear form:

$\frac{dy}{dx} - \frac{y}{x} = 1$

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Checking statement (A): Is it a homogeneous differential equation?

A differential equation is homogeneous if it can be written as:

$\frac{dy}{dx} = f\left(\frac{y}{x}\right)$

where the right side is a function of only $\frac{y}{x}$.

From the equation: $\frac{dy}{dx} = 1 + \frac{y}{x}$

Let $t = \frac{y}{x}$, then:

$\frac{dy}{dx} = 1 + t = f(t)$

Statement (A) is TRUE.

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Checking statement (B): Is it a differential equation of order 2?

The order is the highest derivative present in the equation.

The highest derivative is $\frac{dy}{dx}$ (first derivative only).

This is a first-order equation, not second-order.

Statement (B) is FALSE.

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Checking statement (C): Does the general solution contain 2 arbitrary constants?

A 1st order differential equation has 1 arbitrary constant.

A 2nd order differential equation has 2 arbitrary constants.

Since this equation is 1st order, it will have only 1 arbitrary constant in its general solution.

Statement (C) is FALSE.

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Checking statement (D): Is the integrating factor $\frac{1}{x}$?

The linear form is:

$\frac{dy}{dx} - \frac{y}{x} = 1$

This matches the standard form: $\frac{dy}{dx} + P(x)y = Q(x)$

where $P(x) = -\frac{1}{x}$

Integrating factor:

$I.F. = e^{\int P(x)dx}$

$I.F. = e^{\int -\frac{1}{x}dx}$

$I.F. = e^{-\ln|x|}$

$I.F. = e^{\ln|x|^{-1}}$

$I.F. = \frac{1}{x}$

Statement (D) is TRUE.

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Checking statement (E): Is the degree of the differential equation not defined?

The degree is the power of the highest order derivative (after removing any radicals or fractions involving derivatives).

From: $x\frac{dy}{dx} = x + y$

The highest derivative $\frac{dy}{dx}$ appears to the power of 1.

Therefore, the degree is 1 (well-defined).

Statement (E) is FALSE.

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Only statements (A) and (D) are true.