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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:
(x^4y^4+x^2y^2+xy)ydx+(x^4y^4-x^2y^2+xy)xdy=0
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.
Conclusion
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.