Solving the Differential Equation (x - y f(x, y) - xy f(x - y) 4xy x^2 - y^2)

This article delves into a detailed exploration of a specific differential equation: (x - y f(x, y) - xy f(x - y) 4xy (x^2 - y^2)). The goal is to determine the function (f(x, y)) that satisfies this equation. We will walk through the steps systematically, using variable substitutions and function properties to arrive at a solution.

Step 1: Simplify the Equation

First, let us simplify and interpret the given equation:

(x - y f(x, y) - xy f(x - y) 4xy (x^2 - y^2))

We can rewrite the right-hand side as:

(4xy (x^2 - y^2) 4xy (x y)(x - y))

Now, let's consider the case where (x y).

Step 2: Case (x y)

When (x y), substituting into the equation gives:

(y - y f(y, y) - y^2 f(0) 4y^2 (y^2 - y^2)) → (f(y, y) f(0) 0) → (f(0) 0).

Step 3: General Case (x eq y)

Next, consider the general case where (x eq y). We will divide the entire equation by (x^2 - y^2).

(frac{f(x, y) - xy f(x - y)}{x^2 - y^2} 4xy).

Define (g(x, y) frac{f(x, y) - xy}{xy}).

Then, (g(x, y) - g(x - y) 4xy).

Step 4: Further Substitutions and Properties

Let us substitute (y 0) and (x 0). Note:

(frac{f(x, 0) - xy}{xy} - frac{f(x - 0) - (x - 0)y}{(x - 0)y} 0) → (g(x, 0) - g(x) 0) → (g(x, 0) g(x)).

And, (frac{f(0, y) - xy}{xy} - frac{f(0 - y) - (0 - y)y}{(0 - y)y} 0) → (g(0, y) - g(-y) 0) → (g(0, y) g(-y)). This implies (f) is an odd function when considering (g).

Since (f(x, y)) is continuous and differentiable, we can proceed to the next step.

Step 5: Using Differentiation

Let (x yd) and as (d to 0), we get:

(frac{f(yd, y)}{yd} - frac{f(yd - y)}{yd - y} 4y^2 (y y)(y - y)) → (frac{8y^3}{2y} - f'(0) 4y^2) → (4y^2 - f'(0) 4y^2) → (f'(0) 0).

This simplifies the general form of (f). We can also eliminate (f'(0)) by differentiating once:

(f'(y) yf'(y) - 3y^2) → (f'(y) yf'(y) - 3y^2).

Step 6: General Solution

After substituting (y e^u), we find that the simplest solution is:

(f(x, y) y^3 - yf'(0)). Setting (f'(0) 0) gives:

(f(x, y) y^3).

Therefore, (g(x, y) xy^2) and (f(x, y) xy^3).

Consequently, the solution for (f) is:

(f(x, y) xy^3) and (f(x) x^3).

In summary, the function (f(x, y) xy^3) satisfies the original differential equation (x - y f(x, y) - xy f(x - y) 4xy (x^2 - y^2)).