(x^(2)y^(2)+xy+1)ydx+(x^(2)y^(2)-xy+1)xdy=0

4 min read Jul 03, 2024

Differential Equation: (x^(2)y^(2)+xy+1)ydx+(x^(2)y^(2)-xy+1)xdy=0

In this article, we will explore the differential equation (x^(2)y^(2)+xy+1)ydx+(x^(2)y^(2)-xy+1)xdy=0 and discuss its solution.

What is a Differential Equation?

A differential equation is a mathematical equation that involves an unknown function and its derivatives. It is used to model various phenomena in fields such as physics, engineering, and economics. The solution to a differential equation is a function that satisfies the equation.

The Given Differential Equation

The given differential equation is:

(x^(2)y^(2)+xy+1)ydx + (x^(2)y^(2)-xy+1)xdy = 0

This is a first-order differential equation, meaning that it involves the first derivative of the unknown function.

Separation of Variables

To solve this differential equation, we can use the separation of variables method. This method involves separating the variables x and y and integrating both sides of the equation with respect to one of the variables.

Let's start by rewriting the equation as:

(x^(2)y^(2)+xy+1)dy + (x^(2)y^(2)-xy+1)dx = 0

Now, we can separate the variables by dividing both sides of the equation by (x^(2)y^(2)+xy+1):

dy + (x^(2)y^(2)-xy+1)/(x^(2)y^(2)+xy+1)) dx = 0

Integration

Next, we can integrate both sides of the equation with respect to y:

∫dy + ∫(x^(2)y^(2)-xy+1)/(x^(2)y^(2)+xy+1)) dx = C

where C is the constant of integration.

The first integral is simply y, while the second integral can be evaluated using substitution or partial fractions.

Solution

After integrating and simplifying, we get the general solution to the differential equation:

y = (C - x)/(x^(2) + xy + 1)

where C is the constant of integration.

Conclusion

In this article, we have solved the differential equation (x^(2)y^(2)+xy+1)ydx + (x^(2)y^(2)-xy+1)xdy = 0 using the separation of variables method. The general solution to the equation is y = (C - x)/(x^(2) + xy + 1), where C is the constant of integration.