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Differential Equation: (y-3x^2)dx-x(1-xy^2)dy=0
Introduction
In this article, we will discuss a specific type of differential equation, namely (y-3x^2)dx-x(1-xy^2)dy=0. Differential equations are a fundamental concept in mathematics and are used to model various phenomena in fields such as physics, engineering, and economics.
Definition of a Differential Equation
A differential equation is a mathematical equation that involves an unknown function and its derivatives, and relates the values of the function and its derivatives at different points in space and/or time. In other words, it is an equation that defines a relationship between a function and its derivatives.
The Given Differential Equation
The differential equation we are interested in is (y-3x^2)dx-x(1-xy^2)dy=0. This is a first-order differential equation, meaning it involves only the first derivative of the unknown function.
Geometric Interpretation
Geometrically, the differential equation (y-3x^2)dx-x(1-xy^2)dy=0 represents a family of curves in the xy-plane. Each curve in the family is a solution to the differential equation.
Physical Interpretation
Physically, the differential equation (y-3x^2)dx-x(1-xy^2)dy=0 can be interpreted as a problem of finding the curve that satisfies a certain condition. For example, it could represent the trajectory of a particle moving under the influence of a force field.
Method of Solution
There are several methods to solve differential equations, including separation of variables, integration factor, and numerical methods. In this case, we can use the method of separation of variables to solve the differential equation.
Separation of Variables
The method of separation of variables involves separating the variables x and y on opposite sides of the equation, and then integrating both sides. Let's see how this works for our differential equation.
(y-3x^2)dx-x(1-xy^2)dy=0
Step 1: Separate the variables
Divide both sides of the equation by x to get:
(y/x - 3x)dx = (1-xy^2)dy
Step 2: Integrate both sides
Integrate both sides with respect to x and y, respectively:
∫(y/x - 3x)dx = ∫(1-xy^2)dy
Step 3: Evaluate the integrals
Evaluate the integrals to get:
y ln|x| - x^3/3 = y - xy^3/3 + C
where C is the constant of integration.
General Solution
The general solution to the differential equation (y-3x^2)dx-x(1-xy^2)dy=0 is:
y = (x^3/3 + C) / (ln|x| + xy^2/3)
where C is an arbitrary constant.
Conclusion
In this article, we have discussed the differential equation (y-3x^2)dx-x(1-xy^2)dy=0 and its solution using the method of separation of variables. We have also provided a geometric and physical interpretation of the differential equation. The general solution to the differential equation is y = (x^3/3 + C) / (ln|x| + xy^2/3), where C is an arbitrary constant.
