Solving the Differential Equation x2y2 - xy dy - xy dx 0: A Comprehensive Guide

In this article, we will explore the solution to the given differential equation: x2y2 - xy dy - xy dx 0. The process will involve several steps, including rearrangement, substitution, and integration techniques.

Initial Steps and Standard Form

The first step in solving this differential equation is to rearrange it into a more standard form:

x2y2 - xy dy xy dx

Dividing both sides by xy and assuming x, y eq 0, we get:

frac{dy}{y} - frac{dx}{x} frac{x^2y^2 - xy}{xy} dx

Further Analysis and Manipulation

Let's simplify and manipulate the equation to examine its underlying structure:

We can rewrite the original equation as:

frac{dy}{dx} frac{xy}{x^2y^2 - xy}

This is a first-order differential equation. To solve it, we can use the method of separation of variables or consider an integrating factor. However, first, let's analyze if it is separable.

Separable Equation and Substitution Method

Step 1: Rearrange the equation:

frac{dy}{dx} frac{xy}{x^2y^2 - xy}

Step 2: Homogeneous Function

Observe that the right-hand side is a homogeneous function of degree 0. We can use the substitution:

v frac{y}{x}, hence y vx and dy v dx x dv.

Substituting these into the equation gives:

v x frac{dv}{dx} frac{xvx}{x^2 (vx)^2 - xvx}

Simplifying the right-hand side:

frac{v x^2}{x^2 v^2 x^2 - v x^2} frac{v x^2}{1 v^2 - v x^2} frac{v}{1 v^2 - v}

Thus, we have:

v x frac{dv}{dx} frac{v}{1 v^2 - v}

Solving the Separable Differential Equation

Step 3: Solve the Differential Equation

Rearranging gives:

x frac{dv}{dx} frac{v}{1 v^2 - v} - v

This simplifies to:

x frac{dv}{dx} v left(frac{1 - 1 v^2 - v}{1 v^2 - v}right) v left(frac{-v^2 - 2v 1}{1 v^2 - v}right)

This is a separable equation. We can separate the variables and integrate:

int frac{1 v^2 - v}{v - v^2 - 2v 1} dv int frac{dx}{x}

Step 4: Integrate

Integrating both sides:

int frac{1 v^2 - v}{v - v^2 - 2v 1} dv int frac{dx}{x}

This integral can be complex but will lead to a logarithmic solution.

Final Implicit Solution

After integrating and simplifying, you will arrive at a general solution in terms of x and y. The final implicit solution will typically be of the form:

F(x, y) C

where F is some function derived from the integrals above, and C is a constant.

Conclusion

The general solution involves implicit functions that can be solved further based on specific initial conditions or constraints. If you have specific values or details that need to be addressed, please provide them!