To solve this differential equation, we need to first separate the variables.

Starting with the equation:

4xdx - 3ydy = 3x^ydy - 2xy^dx

Rearranging terms:

4xdx - 3x^ydy - 3ydy + 2xy^dx = 0

Now, we group the terms with dx and dy:

4xdx + 2xy^dx - 3x^ydy - 3ydy = 0

Factoring out the common factors:

x(4dx + 2y^dx) - y(3dx + 3dy) = 0

Now, we can divide both sides by (4dx + 2y^dx) and (3dx + 3dy) to separate the variables:

x/(4 + 2y)dy - y/(3 + 3x)dx = 0

This is now a separated differential equation. The next step would be to integrate both sides with respect to their respective variables.