dy%2Fdx%3D4x%5E3%2B4xy.jpeg "(1-2x^2-2y)dy/dx=4x^3+4xy")
Solving the Differential Equation: (1-2x^2-2y)dy/dx = 4x^3 + 4xy
This article will guide you through the steps involved in solving the given first-order differential equation:
(1-2x^2-2y)dy/dx = 4x^3 + 4xy
This equation is a nonlinear first-order differential equation because it involves terms with both the dependent variable (y) and its derivative (dy/dx).
To solve this equation, we will use the following steps:
1. Rearrange the equation:
We can rewrite the equation as:
dy/dx = (4x^3 + 4xy) / (1-2x^2-2y)
This step makes the equation easier to work with.
2. Identify the type of differential equation:
The equation is a nonlinear first-order differential equation and we can classify it as an exact differential equation. This is because it can be written in the form:
M(x, y) dx + N(x, y) dy = 0
where M(x, y) and N(x, y) are functions of x and y.
In our case:
- M(x, y) = -4x^3 - 4xy
- N(x, y) = 1 - 2x^2 - 2y
We can verify that this is an exact differential equation by checking if:
∂M/∂y = ∂N/∂x
Calculating these partial derivatives:
- ∂M/∂y = -4x
- ∂N/∂x = -4x
Since both partial derivatives are equal, the differential equation is exact.
3. Solve the exact differential equation:
To solve the exact differential equation, we need to find a function u(x, y) such that:
- ∂u/∂x = M(x, y)
- ∂u/∂y = N(x, y)
Integrating M(x, y) with respect to x, we get:
u(x, y) = -x^4 - 2x^2y + g(y)
where g(y) is an arbitrary function of y.
Now, we differentiate this expression with respect to y and set it equal to N(x, y):
∂u/∂y = -2x^2 + g'(y) = 1 - 2x^2 - 2y
This implies that g'(y) = 1 - 2y. Integrating this expression with respect to y, we get:
g(y) = y - y^2 + C
where C is an arbitrary constant.
Therefore, the solution to the exact differential equation is:
u(x, y) = -x^4 - 2x^2y + y - y^2 + C = 0
4. Expressing the solution explicitly:
The solution above is an implicit solution for y in terms of x. It may be possible to express this solution explicitly, depending on the specific form of the equation. However, in this case, it is not straightforward to obtain an explicit solution.
Conclusion:
The solution to the given differential equation is -x^4 - 2x^2y + y - y^2 + C = 0, which represents an implicit solution for y in terms of x. While it may not be possible to express the solution explicitly, this implicit form still provides a valid and complete solution to the given differential equation.