Solving Homogeneous Differential Equations: A Step-by-Step Guide

Homogeneous differential equations are a type of ordinary differential equations (ODE) that can often be solved by transforming them into separable equations. In this guide, we will walk through the solution process with a detailed example and explain key concepts such as integrating factors and separation of variables. Understanding these methods will help you solve similar differential equations efficiently.

Understanding Homogeneous Differential Equations

A homogeneous differential equation is one where the right-hand side is zero, and the left-hand side is a homogeneous function. A function is considered homogeneous if multiplying all variables by a constant results in the same function multiplied by that constant raised to the power of the degree of the equation.

Solving the Equation x - y dx - x dy 0

Consider the given homogeneous differential equation:

x - y dx - x dy 0

Let's follow the steps to solve this equation. First, we can rearrange it to separate the variables:

Rewrite the equation:

x - y dx -x dy

Divide both sides by (x(x - y)):

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

Separate the variables:

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

Integration Process

Now, integrate both sides of the equation:

On the left side:

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

The left side can be solved by substitution or recognizing that the integral is (ln|x - y| C_1), and the right side is (ln|x| C_2).

Combining the results:

(ln|x - y| C_1 -ln|x| C_2)

Simplify the expression:

(ln|x - y| -ln|x| (C_2 - C_1))

Let (C C_2 - C_1), then:

(ln|x - y| -ln|x| C)

(ln|x - y| ln|x| C)

(ln|x^2 - xy| C)

(x^2 - xy e^C)

(x - y frac{k}{x})

where (k e^C)

General Form of the Solution

The general solution to the given differential equation is:

(y x - frac{k}{x})

or, in a more general form, where (C) is an arbitrary constant:

(y x - frac{C}{x})

Verification and Further Exploration

To verify the solution, substitute (y x - frac{C}{x}) back into the original differential equation:

x - ((x - frac{C}{x})) dx - x d((x - frac{C}{x}))

x - (x dx frac{C}{x^2} dx) - x ((1 frac{C}{x^2}) dx)

x - (x dx - frac{C}{x^2} dx) - (x dx - frac{C}{x} dx)

x - (2x dx - frac{C}{x^2} dx)

x - (2x dx - frac{2C}{x^2} dx)

x - (2x dx - frac{2C}{x^2} dx)

x - (x dx - x dy - y dx)

x - (x dx - frac{C}{x^2} dx - frac{C}{x} dx)

x - (x dx - x dy - y dx)

Thus, the solution is verified.

For further exploration, you can use similar techniques to solve other homogeneous differential equations. Understanding the integration factor and separation of variables is crucial in solving such equations.