Solving the Differential Equation x dy - y dx 0: A Comprehensive Guide

Differential equations are fundamental in mathematics, physics, engineering, and many other fields. If you're dealing with the differential equation x dy - y dx 0, this guide will walk you through how to solve it step-by-step. We'll explore various methods and illustrate each step to ensure a clear understanding.

Introduction to the Problem

The given differential equation is:

x dy - y dx 0

This can be rewritten in a more standard form for clarity:

x dy y dx

Step-by-Step Solution

Method 1: Direct Integration

We start by dividing both sides of the equation by (xy), assuming (x eq 0) and (y eq 0):

(frac{dy}{y} frac{dx}{x})

Next, we integrate both sides:

(int frac{dy}{y} int frac{dx}{x})

This integration yields:

(ln|y| ln|x| C)

Where (C) is the constant of integration. To simplify, we exponentiate both sides:

(|y| e^{ln|x| C} e^{ln|x|} cdot e^C |x| cdot e^C)

Lets denote (e^C) as (k), where (k) is a positive constant:

(y kx)

Thus, the general solution of the differential equation is:

(y kx)

Method 2: Using Exact Equations

The original equation (x dy - y dx 0) can be rewritten in a different form:

(frac{xdy - ydx}{x^2} 0)

This simplifies to:

(frac{dleft(frac{y}{x}right)}{dx} 0)

Integrating both sides with respect to (x), we get:

(frac{y}{x} K)

Where (K) is an arbitrary constant. This leads to:

(y kx)

Thus, the same general solution is derived.

Method 3: Using Logarithmic Manipulation

Starting with the equation:

x dy - y dx 0

We rewrite it as:

(frac{dy}{y} frac{dx}{x})

Integrating both sides, we get:

(int frac{dy}{y} - int frac{dx}{x} 0)

This gives:

(ln|y| - ln|x| 0)

Combining the logarithms:

(lnleft|frac{y}{x}right| 0)

Exponentiating both sides:

(left|frac{y}{x}right| 1)

Again, we can denote this as:

(y kx)

Thus, the solution is:

(y kx)

Conclusion

In all the methods discussed above, we arrived at the same general solution: (y kx). Here, (k) is a constant that can be positive or negative. This approach can help you solve similar differential equations by isolating the variables and integrating appropriately.

If you face more complex differential equations, remember to always use logarithms and integration techniques to simplify and solve the problem.