To determine the integrating factor of the given differential equation, we can use the formula:

μ(x) = e^(∫P(x) dx)

where P(x) is the coefficient of dx in the given differential equation.

Given differential equation: 2xy dx - (3x^2 - y^2) dy = 0

1. Identify P(x):
P(x) = 2xy

2. Integrate P(x) with respect to x:
∫P(x) dx = ∫2xy dx = x^2y + C

3. Substitute ∫P(x) dx back into the integrating factor formula:
μ(x) = e^(x^2y + C)

To simplify the integrating factor, we can assume C = 0 for simplicity. This gives:

μ(x) = e^(x^2y)

Now, we need to check if μ(x) satisfies the condition for an integrating factor, which is:

∂(μ(x)Q(x, y))/∂y - ∂(μ(x)P(x, y))/∂x = 0

where Q(x, y) is the coefficient of dy in the given differential equation.

4. Identify Q(x, y):
Q(x, y) = -(3x^2 - y^2)

5. Substitute μ(x) and P(x, y) into the above condition and simplify:

∂(e^(x^2y)(-(3x^2 - y^2)))/∂y - ∂(e^(x^2y)(2xy))/∂x = 0

Differentiating with respect to y:

(e^(x^2y))(2x^2 - 2y) - (x^2)(e^(x^2y))(2xy) = 0

Simplifying the equation:

2x^2e^(x^2y) - 2ye^(x^2y) - 2x^3ye^(x^2y) = 0

e^(x^2y)(2x^2 - 2y - 2x^3y) = 0

The equation holds true if and only if (2x^2 - 2y - 2x^3y) = 0.

6. Solve the equation for y:

2x^2 - 2y - 2x^3y = 0

Rearranging the terms:

2x^2 - 2x^3y = 2y

2x^2(1 - x^3y) = 2y

Dividing both sides by 2:

x^2(1 - x^3y) = y

Expanding the equation:

x^2 - x^5y = y

Rearranging the terms:

x^2 - y = x^5y

Simplifying further:

x^2 - y = x^5y

x^2 = (x^5 + 1)y

y = x^2 / (x^5 + 1)

7. Substitute the value of y back into the equation (2x^2 - 2y - 2x^3y) = 0:

2x^2 - 2(x^2 / (x^5 + 1)) - 2x^3(x