dx-2xy-dy%3D0.jpeg "(x2+y2)dx-2xy Dy=0")
Differential Equation: (x²+y²)dx - 2xy dy = 0
Introduction
In this article, we will discuss the differential equation (x²+y²)dx - 2xy dy = 0. This is a first-order differential equation that involves the variables x and y. We will explore the solution to this equation and discuss its properties.
Separation of Variables
To solve this differential equation, we can use the method of separation of variables. This involves separating the variables x and y and integrating both sides of the equation.
Rearranging the equation, we get:
$\frac{dy}{dx} = \frac{x²+y²}{2xy}$
Now, we can separate the variables:
$\int \frac{2xy}{x²+y²} dy = \int dx$
Solving the Equation
To solve the equation, we need to evaluate the integrals on both sides.
$\int \frac{2xy}{x²+y²} dy = \int dx$
$\Rightarrow \ln(x²+y²) = x + C$
where C is the constant of integration.
Properties of the Solution
The solution to the differential equation has several important properties:
- Symmetry: The solution is symmetric about the origin, meaning that if (x, y) is a solution, then (-x, -y) is also a solution.
- Homogeneity: The solution is homogeneous, meaning that if (x, y) is a solution, then (kx, ky) is also a solution for any constant k.
Conclusion
In this article, we have solved the differential equation (x²+y²)dx - 2xy dy = 0 using the method of separation of variables. We have also discussed the properties of the solution, including symmetry and homogeneity. This differential equation has applications in various fields, including physics, engineering, and economics.
