dy%2Fdx%3D4x%5E3%2B4xy.jpeg "(1-2x^2-2y)dy/dx=4x^3+4xy")
Differential Equation: (1-2x^2-2y)dy/dx=4x^3+4xy
In this article, we will discuss the solution to the differential equation (1-2x^2-2y)dy/dx=4x^3+4xy.
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:
(1-2x^2-2y)dy/dx=4x^3+4xy
This is a first-order ordinary differential equation, where the derivative of y with respect to x is represented by dy/dx.
Solving the Differential Equation
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.
First, we can rewrite the equation as:
dy/dx=(4x^3+4xy)/(1-2x^2-2y)
Next, we can separate the variables by moving the dx term to the right-hand side:
∫(1-2x^2-2y)^(-1) dy = ∫(4x^3+4xy) dx
Now, we can integrate both sides of the equation:
∫(1-2x^2-2y)^(-1) dy = x^4 + 2x^2y + C
where C is the constant of integration.
Simplifying the Solution
To simplify the solution, we can write it in the form:
y = (x^4 + 2x^2y + C)/(1-2x^2)
This is the general solution to the differential equation.
Conclusion
In this article, we have solved the differential equation (1-2x^2-2y)dy/dx=4x^3+4xy using the method of separation of variables. The general solution to the equation is y = (x^4 + 2x^2y + C)/(1-2x^2), where C is the constant of integration.
