General Integral of the PDE px(x y) =qy(x y) -(x-y) (2x 2y z)

The given partial differential equation is px(x y) = qy(x y) - (x - y)(2x 2y z). To find the general integral of this PDE, we need to follow the following steps:

Step 1: Find the Integrating Factor

We need to multiply both sides of the equation by an integrating factor, which is defined as follows:

Integrating Factor (IF) = e^(integral(p(x) dx))

Here, p(x) is the coefficient of the partial derivative with respect to x. In this case, p(x) = p(x,y) = p(x).

So, the integrating factor for our PDE is:

IF = e^(integral(p(x) dx)) = e^(px dx)

Step 2: Multiply Both Sides of the Equation by the Integrating Factor

Multiplying both sides of the equation by the integrating factor, we get:

(px e^(px) + Qy e^(px))dx - (2x^2 yz - xy^2) e^(px) dy = 0

Step 3: Integrate Both Sides of the Equation

Integrating both sides of the equation, we get:

∫(px e^(px) + Qy e^(px))dx - ∫(2x^2 yz - xy^2) e^(px) dy = C

Here, C is the constant of integration.

Step 4: Simplify the Equation

Simplifying the equation, we get:

x e^(px) Q(y) = ∫(2x^2 yz - xy^2) e^(px) dy + C

Here, Q(y) is the integrating factor with respect to y.

Step 5: Solve for the General Integral

To solve for the general integral, we need to integrate the right-hand side of the equation with respect to y. This can be done by using integration by parts or any other suitable method.

Once we have the general integral, we can substitute any initial or boundary conditions to find the particular solution.

Therefore, the general integral of the given PDE is:

x e^(px) Q(y) = ∫(2x^2 yz - xy^2) e^(px) dy + C

where Q(y) is the integrating factor with respect to y and C is the constant of integration.