Singular Integral of Partial Differential Equation

To find the singular integral of the given partial differential equation, we need to follow certain steps. Let's break down the process into detailed steps:

Step 1: Define the Partial Differential Equation (PDE)

The given PDE is z = px*qy*log(pq), where p and q are the independent variables, and z is the dependent variable.

Step 2: Identify the Singular Integral

In order to find the singular integral of a PDE, we need to determine the integral of the given PDE with respect to one of the variables, while treating the other variable as a constant. In this case, we can choose to integrate with respect to either p or q.

Step 3: Choose the Variable to Integrate

Let's choose to integrate with respect to p. Therefore, we will treat q as a constant and differentiate z with respect to p.

Step 4: Calculate the Partial Derivative

Taking the partial derivative of z with respect to p, we get:

∂z/∂p = x*qy*log(pq) + px*qy/p

Simplifying further, we have:

∂z/∂p = x*qy*log(pq) + qy

Step 5: Integrate the Partial Derivative

Now, we integrate the partial derivative with respect to p:

∫ (∂z/∂p) dp = ∫ (x*qy*log(pq) + qy) dp

This integration will result in a function of p and q, which we can denote as F(p, q).

Step 6: Determine the Singular Integral

The singular integral of the PDE is given by:

∫ F(p, q) dq + g(q)

Here, g(q) represents the arbitrary function of q.

Step 7: Finalize the Singular Integral

Combining the previous steps, the singular integral of the given PDE can be expressed as:

∫ (∫ (x*qy*log(pq) + qy) dp) dq + g(q)

This is the final form of the singular integral of the given PDE.