For the differential equation $(x + y)dy + (x - y)dx = 0$, which of the following is/are correct?

(A) Differential equation is homogeneous

(B) Order of differential equation is 1

(C) Integrating factor of differential equation is $e^x$

(D) Degree of the equation is not defined

Choose the correct answer from the options given below:

The equation can be rewritten as $M\,dx + N\,dy = 0$:

$(x - y)\,dx + (x + y)\,dy = 0$

where $M(x, y) = x - y$ and $N(x, y) = x + y$.

Replacing $x$ with $tx$ and $y$ with $ty$:

$M(tx, ty) = tx - ty = t(x - y) = t^1 \cdot M(x,y)$

$N(tx, ty) = tx + ty = t(x + y) = t^1 \cdot N(x,y)$

Both are homogeneous functions of degree 1.

(A) is correct.

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The equation can be written as:

$\dfrac{dy}{dx} = \dfrac{y - x}{x + y}$

The highest derivative present is $\dfrac{dy}{dx}$, which is a first-order derivative.

(B) is correct.

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Checking whether the equation is exact:

$\dfrac{\partial M}{\partial y} = \dfrac{\partial(x - y)}{\partial y} = -1$

$\dfrac{\partial N}{\partial x} = \dfrac{\partial(x + y)}{\partial x} = +1$

Since $-1 \neq 1$, the equation is not exact.

Testing $\mu = e^x$ as an integrating factor:

$\mu M = (x - y)e^x, \qquad \mu N = (x + y)e^x$

$\dfrac{\partial(\mu M)}{\partial y} = -e^x$

$\dfrac{\partial(\mu N)}{\partial x} = e^x + (x + y)e^x = (1 + x + y)e^x$

$-e^x \neq (1 + x + y)e^x$

The equation does not become exact, so $e^x$ is not a valid integrating factor.

(C) is incorrect.

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The equation:

$(x + y)\dfrac{dy}{dx} + (x - y) = 0$

Here $\dfrac{dy}{dx}$ appears as a polynomial of degree 1 with no transcendental functions involved. The degree is well-defined and equal to 1.

(D) is incorrect.

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Statements (A) and (B) are correct.

$\boxed{\textbf{Option 4: (A) and (B) only}}$