Solving the Differential Equation (6x) dx + dy = 0

In this article, we will explore the process of solving the given differential equation: (6x) dx + dy = 0. This type of differential equation can be solved using the method of separation of variables, and we will introduce the arbitrary constant C in the solution.

Step 1: Rearrange the Equation

The first step is to rearrange the given differential equation to isolate the variables on different sides of the equation. In this case, we have:

(6x) dx + dy = 0

Dividing both sides by (6x), we get:

dx/x + (1/6) dy = 0

Step 2: Separate the Variables

Now, we can separate the variables by moving all the x-related terms to one side and all the y-related terms to the other side:

dx/x = -(1/6) dy

Step 3: Integrate Both Sides

To solve the differential equation, we need to integrate both sides of the equation:

∫ dx/x = ∫ -(1/6) dy ln|x| = -(1/6) y + C

Step 4: Solve for y

Rearranging the equation, we get:

y = -6 ln|x| + C

where C is the arbitrary constant introduced during the integration process.

Conclusion

In this article, we have successfully solved the given differential equation (6x) dx + dy = 0 using the method of separation of variables and introducing the arbitrary constant C. The final solution is y = -6 ln|x| + C, which represents the family of solutions to the differential equation.

Remember that the arbitrary constant C can take any real value, allowing for different specific solutions to the differential equation based on the given initial conditions or boundary conditions.