Solving the Differential Equation 1x dy - y dx 0: A Step-by-Step Guide

In the realm of differential equations, the equation 1x dy - y dx 0 is a classic example of a separable equation. This article will walk you through the process of solving this equation, providing a thorough understanding of the methods involved. Whether you are a student, a mathematician, or anyone interested in differential equations, this guide will help you master this essential skill.

Understanding the Equation

The given differential equation is:

1x dy - y dx 0

This is a first-order ordinary differential equation (ODE). The goal is to find the function y(x) that satisfies this equation.

Separation of Variables

The first step in solving a separable differential equation is to separate the variables. In the given equation 1x dy - y dx 0, we can rewrite it as:

1x dy y dx

Next, we divide both sides of the equation by y and x:

dy/y dx/x

This step separates the variables, with all terms involving y on one side and all terms involving x on the other.

Integration

The next step is to integrate both sides of the equation. Integrating the left side with respect to y and the right side with respect to x, we obtain:

∫(1/y) dy ∫(1/x) dx

This integration yields the natural logarithms of both variables:

ln|y| ln|x| C

where C is the constant of integration. Here's a detailed look at the integration:

Assuming we can combine the constants of integration (Cy and Cx) into a single constant C:

ln|y| ln|x| C

Simplifying the Solution

To simplify this result, we can use properties of logarithms. Specifically, we can rewrite the equation as:

ln|y| - ln|x| C

Using the logarithm subtraction rule, we have:

ln|y/x| C

Exponentiating both sides to remove the logarithm, we get:

y/x eC

Let's denote eC as another constant C (for simplicity), so the solution becomes:

y Cx

This is the general solution to the differential equation 1x dy - y dx 0.

Conclusion

In summary, the differential equation 1x dy - y dx 0 can be solved using the method of separation of variables and integration. The final solution is:

y Cx

Understanding and mastering this process not only helps in solving similar equations but also builds a solid foundation in differential equations. Whether you are applying this knowledge in pure mathematics or real-world applications, such as physics and engineering, this method proves to be a powerful tool.