Given differential equation:
2xy dx + (y^2 - x^2) dy = 0

Explanation:
The given differential equation is not in the standard form of a first-order linear differential equation. In order to solve it, we need to find an integrating factor that will help us transform the equation into a linear form.

Step 1: Identify the coefficient functions
In the given equation, we have two coefficient functions: M(x, y) = 2xy and N(x, y) = y^2 - x^2.

Step 2: Check for exactness
To check whether the equation is exact, we compute the partial derivatives of M and N with respect to y and x respectively:

∂M/∂y = 2x
∂N/∂x = -2x

Since ∂M/∂y is not equal to ∂N/∂x, the equation is not exact.

Step 3: Find the integrating factor
To find the integrating factor, we divide the equation by the coefficient function of dy (N(x, y)):

(2xy dx + (y^2 - x^2) dy) / (y^2 - x^2) = 0

Simplifying the above equation:

2xy/y^2 - x^2 dx + dy = 0
(2x/y - x^2/y^2) dx + dy = 0

Now, we compare the coefficients of dx and dy with the general form of a linear differential equation:

M(x, y) dx + N(x, y) dy = 0

Comparing the coefficients, we get:

M(x, y) = 2x/y - x^2/y^2
N(x, y) = 1

Step 4: Calculate the integrating factor
The integrating factor (IF) is given by the formula:

IF = e^(∫(∂M/∂y - ∂N/∂x) dx)

In this case, (∂M/∂y - ∂N/∂x) = (2x - (-2x)) = 4x.

Therefore, the integrating factor is:

IF = e^(∫4x dx) = e^(2x^2)

Step 5: Multiply the equation by the integrating factor
Multiplying the original equation by the integrating factor (e^(2x^2)):

e^(2x^2) (2xy dx + (y^2 - x^2) dy) = 0

Simplifying the equation:

2xye^(2x^2) dx + (y^2 - x^2)e^(2x^2) dy = 0

This equation is now in the form of a linear differential equation, which can be solved using standard techniques.

Therefore, the correct answer is option 'C' (1/y^2).