3 min read Jun 17, 2024

Solving the Differential Equation: (x^4y^4+x^2y^2+xy)ydx+(x^4y^4-x^2y^2+xy)xdy=0

This article explores the solution process for the given differential equation:


This equation is a first-order homogeneous differential equation. We can solve it using the following steps:

1. Rearranging the Equation

First, let's rearrange the equation to make it easier to work with. Divide both sides by xy:

(x^3y^3 + xy + 1)ydx + (x^3y^3 - xy + 1)xdy = 0

Now, divide both sides by (x^3y^3 + xy + 1)(x^3y^3 - xy + 1):

(ydx)/(x^3y^3 - xy + 1) + (xdy)/(x^3y^3 + xy + 1) = 0

2. Substitution and Integration

Let's make the following substitution:

u = xy

Then, du = xdy + ydx. Substituting these into our rearranged equation gives:

(du)/(u^3 - u + 1) = 0

This equation is now separable. Integrating both sides, we get:

∫(du)/(u^3 - u + 1) = ∫0 du

The integral on the left-hand side can be solved using partial fractions. However, the exact solution involves complex numbers and is quite intricate.

3. General Solution

After solving the integral on the left-hand side, we'll obtain a function of u equal to a constant. Substituting back u = xy gives us the general solution of the differential equation in terms of x and y.


While the complete solution involves complex integrals, the process outlined above provides a general framework for solving the given differential equation. The key steps involve rearranging the equation, making a suitable substitution, and integrating both sides. This method can be applied to similar homogeneous differential equations.

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