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

4 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, which is a first-order, non-linear differential equation. We will employ methods of separation of variables and integration to arrive at a general solution.

1. Rearranging the Equation

First, we rearrange the equation to separate the terms involving 'x' and 'y':

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

Next, we separate the variables by dividing both sides by (x - e^(arctan(y))) and multiplying both sides by dx:

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

2. Integrating Both Sides

Now, we integrate both sides of the equation:

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

The left-hand side integrates directly to arctan(y):

arctan(y) = -∫ dx / (x - e^(arctan(y)))

The right-hand side requires a substitution to simplify the integral. Let u = x - e^(arctan(y)). Then, du = dx. Substituting these values, we get:

arctan(y) = -∫ du / u

This integral evaluates to -ln|u| + C, where C is the constant of integration:

arctan(y) = -ln|x - e^(arctan(y))| + C

3. Solving for y

To obtain an explicit solution for y, we need to isolate it. First, we take the exponential of both sides:

e^(arctan(y)) = e^(-ln|x - e^(arctan(y))| + C)

Simplifying the right-hand side:

e^(arctan(y)) = e^C / |x - e^(arctan(y))|

Let K = e^C. We can write:

e^(arctan(y)) = K / |x - e^(arctan(y))|

Now, we can solve for y. We can either leave the solution in this implicit form or further manipulate it to obtain an explicit solution for y. However, due to the complexity of the equation, obtaining a fully explicit solution for y might not be possible.

Conclusion

The given differential equation can be solved using separation of variables and integration, leading to an implicit solution for y. While it might not be possible to obtain a fully explicit solution for y, the implicit solution represents the general solution of the given differential equation. This solution provides valuable information about the relationship between x and y and can be used for further analysis.