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

5 min read Jun 16, 2024

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

This article explores the solution to the given differential equation. We'll utilize techniques of exact differential equations to arrive at the general solution.

1. Identifying the Equation as Exact

The given differential equation is of the form:

M(x, y) + N(x, y) dy/dx = 0

Where:

M(x, y) = 1 + y^2
N(x, y) = x - e^(arctan(y))

A differential equation is considered exact if:

∂M/∂y = ∂N/∂x

Let's check if our equation satisfies this condition:

∂M/∂y = 2y
∂N/∂x = 1

Since ∂M/∂y ≠ ∂N/∂x, the given equation is not exact.

2. Finding an Integrating Factor

To make the equation exact, we can multiply it by an integrating factor, μ(x, y). The integrating factor is chosen such that the following condition holds:

(μM)_y = (μN)_x

This can be simplified to:

μ(∂M/∂y - ∂N/∂x) = N(∂μ/∂x) - M(∂μ/∂y)

We notice that the left-hand side depends only on y, while the right-hand side depends only on x. This suggests we should look for an integrating factor that is either a function of x only (μ(x)) or a function of y only (μ(y)).

Let's assume μ is a function of x only:

(μ(x)) (∂M/∂y - ∂N/∂x) = N(dμ/dx)

Substituting the values of M, N, and their derivatives:

μ(x) (2y - 1) = (x - e^(arctan(y))) (dμ/dx)

Separating the variables and integrating:

∫(2y-1)/N dy = ∫1/μ dμ

This integral may not be straightforward to solve, suggesting we consider an integrating factor that's a function of y only.

Let's assume μ is a function of y only:

(μ(y)) (∂M/∂y - ∂N/∂x) = -M(dμ/dy)

Substituting the values of M, N, and their derivatives:

μ(y) (2y - 1) = -(1 + y^2)(dμ/dy)

Separating the variables and integrating:

∫(1 + y^2)/(2y-1) dy = -∫1/μ dμ

This integral is easier to solve. The integral on the left can be evaluated using partial fractions, and the integral on the right results in -ln(μ). Solving for μ(y), we obtain:

μ(y) = 1/(1 + y^2)

3. Solving the Exact Equation

Now, we multiply the original equation by the integrating factor:

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

This simplifies to:

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

This equation is now exact. We can find a solution by integrating the following:

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

Solving the integrals:

x + ∫(x - e^(arctan(y)))/(1 + y^2) dy = C

The integral involving y can be solved using substitution (u = arctan(y)). The solution is:

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

4. General Solution

The general solution to the differential equation is:

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

where C is an arbitrary constant.