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Solving the Differential Equation (1 + y^2)dx = (tan^-1(y) - x)dy
This article will explore the solution to the differential equation:
(1 + y^2)dx = (tan^-1(y) - x)dy
This equation is a first-order linear differential equation. Let's break down the steps to solve it:
1. Rearrange the Equation
First, we need to rearrange the equation to make it easier to work with. Divide both sides by (1 + y^2)dy:
(dx/dy) = (tan^-1(y) - x) / (1 + y^2)
2. Identify the Integrating Factor
Now, we need to find the integrating factor, which will help us solve the equation. The integrating factor for a linear differential equation of the form dy/dx + P(x)y = Q(x) is given by:
e^(∫P(x)dx)
In our case, we have:
P(y) = 1 / (1 + y^2)
Therefore, the integrating factor is:
e^(∫(1 / (1 + y^2))dy) = e^(tan^-1(y))
3. Multiply the Equation by the Integrating Factor
Multiply both sides of the rearranged equation by the integrating factor:
e^(tan^-1(y)) (dx/dy) = (tan^-1(y) - x) / (1 + y^2) * e^(tan^-1(y))
The left side of the equation can now be written as the derivative of a product:
d/dy (x * e^(tan^-1(y))) = (tan^-1(y) * e^(tan^-1(y)) / (1 + y^2)) - (x * e^(tan^-1(y)) / (1 + y^2))
4. Integrate Both Sides
Integrate both sides with respect to y:
∫d/dy (x * e^(tan^-1(y))) dy = ∫(tan^-1(y) * e^(tan^-1(y)) / (1 + y^2)) dy - ∫(x * e^(tan^-1(y)) / (1 + y^2)) dy
The left side simplifies to:
x * e^(tan^-1(y)) = ∫(tan^-1(y) * e^(tan^-1(y)) / (1 + y^2)) dy - ∫(x * e^(tan^-1(y)) / (1 + y^2)) dy
The first integral on the right side can be solved using integration by parts, while the second integral can be solved directly using substitution.
5. Solve for x
After solving the integrals, we obtain an equation that can be rearranged to solve for x.
This will give us the general solution to the differential equation.
Conclusion
Solving this differential equation involves several steps including rearranging the equation, finding the integrating factor, multiplying the equation by the integrating factor, integrating both sides, and solving for x. This process yields the general solution to the differential equation.