(1+x^4)dy+x(1+4y^2)dx=0

4 min read Jun 16, 2024

Solving the Differential Equation: (1+x^4)dy + x(1+4y^2)dx = 0

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

(1 + x^4)dy + x(1 + 4y^2)dx = 0

This equation is a first-order, nonlinear differential equation. We will use the method of separation of variables to solve it.

1. Rearranging the Equation

First, we need to rearrange the equation to separate the variables 'x' and 'y' on different sides of the equation:

(1 + x^4)dy = -x(1 + 4y^2)dx

Now, divide both sides by (1 + x^4) and (1 + 4y^2):

dy / (1 + 4y^2) = -xdx / (1 + x^4)

2. Integrating Both Sides

Now we have successfully separated the variables. The next step is to integrate both sides of the equation:

∫ dy / (1 + 4y^2) = -∫ xdx / (1 + x^4)

To solve these integrals, we can use the following substitutions:

For the left side: Let u = 2y, then du = 2dy.
For the right side: Let v = x^2, then dv = 2xdx.

Substituting these into the integrals, we get:

(1/2) ∫ du / (1 + u^2) = -(1/2) ∫ dv / (1 + v^2)

Now, we can directly integrate both sides:

(1/2) arctan(u) = -(1/2) arctan(v) + C

where C is the constant of integration.

3. Replacing Substitutions

Let's replace 'u' and 'v' with their original values:

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

4. Solving for y

Finally, we can solve for 'y' to obtain the general solution:

arctan(2y) = -arctan(x^2) + 2C

2y = tan(-arctan(x^2) + 2C)

y = (1/2) * tan(-arctan(x^2) + 2C)

This is the general solution of the given differential equation.

Conclusion

We have successfully solved the differential equation (1 + x^4)dy + x(1 + 4y^2)dx = 0 using the method of separation of variables. The final solution is given by: