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Solving the Differential Equation: (1+y^(2))tan^(-1)x dx+2y(1+x^(2))dy=0
This article will explore the solution process for the given differential equation:
(1+y^(2))tan^(-1)x dx+2y(1+x^(2))dy=0
This equation is a first-order, non-linear differential equation. Let's break down the steps to solve it:
1. Identifying the Type of Differential Equation
The given equation is an exact differential equation. This means it can be expressed as the total differential of some function F(x,y). To verify this, we check if the following condition holds:
∂M/∂y = ∂N/∂x
Where:
- M(x,y) = (1+y^(2))tan^(-1)x
- N(x,y) = 2y(1+x^(2))
Let's calculate the partial derivatives:
- ∂M/∂y = 2y
- ∂N/∂x = 2y
Since ∂M/∂y = ∂N/∂x, the equation is indeed exact.
2. Finding the Potential Function F(x,y)
To find F(x,y), we integrate M(x,y) with respect to x, treating y as a constant:
F(x,y) = ∫(1+y^(2))tan^(-1)x dx = (1+y^(2))∫tan^(-1)x dx + g(y)
Where g(y) is an arbitrary function of y.
We know that the integral of tan^(-1)x dx is:
x tan^(-1)x - 1/2 ln(1+x^(2))
Therefore,
F(x,y) = (1+y^(2))(x tan^(-1)x - 1/2 ln(1+x^(2))) + g(y)
Now, we differentiate F(x,y) with respect to y and equate it to N(x,y):
∂F/∂y = 2y(x tan^(-1)x - 1/2 ln(1+x^(2))) + g'(y) = 2y(1+x^(2))
Solving for g'(y):
g'(y) = 2y(1+x^(2)) - 2y(x tan^(-1)x - 1/2 ln(1+x^(2))) = 2y + y ln(1+x^(2))
Integrating g'(y) with respect to y:
g(y) = y^(2) + 1/2 y^(2) ln(1+x^(2)) + C
Where C is the constant of integration.
Finally, we substitute g(y) back into F(x,y):
F(x,y) = (1+y^(2))(x tan^(-1)x - 1/2 ln(1+x^(2))) + y^(2) + 1/2 y^(2) ln(1+x^(2)) + C
3. The Implicit Solution
The implicit solution to the differential equation is given by:
F(x,y) = C
Therefore, the solution is:
(1+y^(2))(x tan^(-1)x - 1/2 ln(1+x^(2))) + y^(2) + 1/2 y^(2) ln(1+x^(2)) = C
Conclusion
This article demonstrated how to solve the given differential equation by recognizing it as an exact equation. We found the potential function F(x,y) and obtained the implicit solution. This process is a powerful tool for solving a wide range of differential equations.