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Differential Equation: (1+y^2)dx + (1+x^2)dy = 0
Introduction
In mathematics, a differential equation is an equation that involves an unknown function and its derivatives. In this article, we will explore a specific type of differential equation: (1+y^2)dx + (1+x^2)dy = 0.
What is a Differential Equation?
A differential equation is a mathematical equation that involves an unknown function and its derivatives. It is a fundamental concept in various fields, including physics, engineering, and economics. Differential equations are used to model various phenomena, such as population growth, chemical reactions, and electrical circuits.
The Equation: (1+y^2)dx + (1+x^2)dy = 0
The equation (1+y^2)dx + (1+x^2)dy = 0 is a first-order ordinary differential equation. It is a separable equation, which means that it can be solved by separating the variables and integrating both sides.
Solution to the Equation
To solve this equation, we can start by rewriting it in the form:
dy/dx = -(1+y^2) / (1+x^2)
This is a separable equation, and we can solve it by separating the variables and integrating both sides.
First, we can integrate the left-hand side with respect to y:
∫(1+y^2) dy = ∫(1+x^2) dx
Evaluating the integrals, we get:
y + (y^3)/3 = x + (x^3)/3 + C
where C is the constant of integration.
Interpretation of the Solution
The solution to the equation (1+y^2)dx + (1+x^2)dy = 0 is a family of curves, each corresponding to a different value of the constant C. These curves can be plotted on a graph, and they can be used to model various physical phenomena.
For example, in physics, this equation can be used to model the motion of an object in a gravitational field. The solution to the equation can be used to calculate the trajectory of the object.
Conclusion
In conclusion, the equation (1+y^2)dx + (1+x^2)dy = 0 is a first-order ordinary differential equation that can be solved using separation of variables. The solution to the equation is a family of curves, each corresponding to a different value of the constant C. This equation has many applications in physics, engineering, and other fields, and it is an important tool for modeling various phenomena.
