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Differential Equation: (3x^2+6xy^2)dx+(6x^2y+4y^2)dy=0
In this article, we will discuss the differential equation (3x^2+6xy^2)dx+(6x^2y+4y^2)dy=0. We will explore the characteristics of this equation, its solution, and the methods used to solve it.
Definition and Formulation
A differential equation is an equation that involves an unknown function and its derivatives. In this case, we have a first-order differential equation, which means that it involves the first derivative of the unknown function.
The given differential equation is:
(3x^2+6xy^2)dx+(6x^2y+4y^2)dy=0
This equation is a nonlinear differential equation, meaning that it cannot be written in the form of a linear combination of the independent variable and its derivatives.
Separation of Variables
To solve this differential equation, we can use the method of separation of variables. This method involves separating the variables x and y, and then integrating both sides of the equation separately.
First, we can rewrite the equation as:
(3x^2+6xy^2)dx = -(6x^2y+4y^2)dy
Next, we can separate the variables by dividing both sides of the equation by (3x^2+6xy^2):
dx = -((6x^2y+4y^2)/(3x^2+6xy^2))dy
Now, we can integrate both sides of the equation with respect to y:
∫(1)dx = -∫((6x^2y+4y^2)/(3x^2+6xy^2))dy
Solution
The solution to the differential equation is:
x = F(y) - (2y^2)/3
where F(y) is the constant of integration.
Conclusion
In this article, we have discussed the differential equation (3x^2+6xy^2)dx+(6x^2y+4y^2)dy=0. We have seen how to use the method of separation of variables to solve this nonlinear differential equation. The solution involves integrating both sides of the equation separately and then solving for x in terms of y. The final solution is x = F(y) - (2y^2)/3, where F(y) is the constant of integration.
