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(1+y^2)+(x-e^tan^-1x)dy/dx=0

7 min read Jun 16, 2024
(1+y^2)+(x-e^tan^-1x)dy/dx=0

Solving the Differential Equation: (1 + y^2) + (x - e^(arctan(x))) dy/dx = 0

This article will guide you through the process of solving the given differential equation:

(1 + y^2) + (x - e^(arctan(x))) dy/dx = 0

Understanding the Equation

This is a first-order differential equation. It is non-linear due to the presence of the y^2 term. To solve this equation, we can utilize the method of exact differential equations.

Steps to Solve

  1. Check for Exactness:

    • A differential equation of the form M(x,y)dx + N(x,y)dy = 0 is considered exact if ∂M/∂y = ∂N/∂x.
    • Rearrange the given equation to the standard form: (x - e^(arctan(x))) dy/dx = -(1 + y^2) (x - e^(arctan(x))) dy + (1 + y^2) dx = 0
    • Identify M(x,y) = (1 + y^2) and N(x,y) = (x - e^(arctan(x))).
    • Calculate the partial derivatives:
      • ∂M/∂y = 2y
      • ∂N/∂x = 1 - (1/(1 + x^2)) * e^(arctan(x))
    • Since ∂M/∂y ≠ ∂N/∂x, the given equation is not exact.

  2. Finding an Integrating Factor:

    • Since the equation is not exact, we need to find an integrating factor, μ(x,y), to make it exact.
    • If (∂M/∂y - ∂N/∂x)/N is a function of x alone, then μ(x) = exp(∫((∂M/∂y - ∂N/∂x)/N) dx) is an integrating factor.
    • In this case, ((∂M/∂y - ∂N/∂x)/N) = (2y - (1 - (1/(1 + x^2)) * e^(arctan(x)))) / (x - e^(arctan(x))) is not a function of x alone.
    • If (∂N/∂x - ∂M/∂y)/M is a function of y alone, then μ(y) = exp(∫((∂N/∂x - ∂M/∂y)/M) dy) is an integrating factor.
    • Let's calculate: ((∂N/∂x - ∂M/∂y)/M) = ((1 - (1/(1 + x^2)) * e^(arctan(x))) - 2y) / (1 + y^2). This expression simplifies to -2y / (1 + y^2), a function of y alone.
    • Therefore, our integrating factor is μ(y) = exp(∫(-2y/(1 + y^2)) dy) = exp(-ln(1 + y^2)) = 1/(1 + y^2).

  3. Multiplying by the Integrating Factor:

    • Multiply the original equation by the integrating factor: (1/(1 + y^2))((x - e^(arctan(x))) dy + (1 + y^2) dx) = 0 (x - e^(arctan(x)))/(1 + y^2) dy + dx = 0

  4. Verifying Exactness:

    • Now, let's verify if the equation is exact:
      • M(x,y) = 1, N(x,y) = (x - e^(arctan(x)))/(1 + y^2).
      • ∂M/∂y = 0, ∂N/∂x = 1/(1 + y^2) - (1/(1 + x^2)) * e^(arctan(x))/(1 + y^2) = 0.
    • Since ∂M/∂y = ∂N/∂x, the equation is now exact.

  5. Finding the Solution:

    • An exact differential equation can be written as dF(x,y) = 0, where F(x,y) is a potential function.
    • We need to find F(x,y) such that:
      • ∂F/∂x = M(x,y) = 1
      • ∂F/∂y = N(x,y) = (x - e^(arctan(x)))/(1 + y^2)
    • Integrating the first equation with respect to x, we get: F(x,y) = x + g(y), where g(y) is an arbitrary function of y.
    • Differentiating this expression with respect to y and equating it to N(x,y), we obtain: g'(y) = (x - e^(arctan(x)))/(1 + y^2).
    • Integrating this equation with respect to y, we get: g(y) = arctan(y) - e^(arctan(x)) * arctan(y) + C.
    • Finally, the general solution of the differential equation is: F(x,y) = x + arctan(y) - e^(arctan(x)) * arctan(y) + C = 0, where C is an arbitrary constant.

The General Solution

The general solution of the given differential equation is:

x + arctan(y) - e^(arctan(x)) * arctan(y) = C

This equation represents a family of curves, with each curve corresponding to a specific value of the constant C.

Important Note:

This solution is implicit, meaning it is not expressed explicitly in the form y = f(x). Depending on the context and specific requirements, you might need to use numerical methods or further manipulations to obtain an explicit solution or analyze the behavior of the solution curves.

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