Homogeneity Differential Equations

May 28, 2024

In differential equations, the term homogeneity signifies a particular kind of structure that renders some classes of differential equations amenable to certain methods of solution. Non-homogeneous differential equations are those which can not be expressed in the form where each of the given terms is a function of a homogeneous form of equal degree. This characteristic allows for the use of a method of substitution to solve them in the simplest way possible.

What is the Homogeneity Differential Equation and Its Characteristics?

This first-order differential equation {dy}÷{dx} = f(x, y) is said to be homogeneous as the function f(x, y) depends only on ({y}÷{x}). Mathematically, f(x, y) is homogeneous of degree zero if:

f(tx, ty) = f(x, y)

For all non-zero t. This means that scaling x and y by the same factor does not change the value of the function. An example of such an equation is:

{dy}÷{dx} = {ax + by}÷{cx + dy}

Solving Homogeneous Differential Equations

In solving a homogeneous differential equation, the usual method adopted involves making a substitution that allows the equation to be put in the form of a separable equation. Covariance is the most common substitution (v={y}÷{x})where (y= vx). By substituting (y= vx) and differentiating with respect to (x), we obtain:

{dy}÷{dx} = v + x{dv}÷{dx}

Substituting these into the original differential equation {dy}÷{dx} = f(x, y) we get:

[v + x{dv}÷{dx} = f(x, vx)]

Since f(x, y) is homogeneous, f(x, vx) can be written in terms of (v). Thus, the equation reduces to:

[v + x{dv}÷{dx} = g(v)]

This equation is separable and can be written as:

{dv}÷{g(v) – v} = {dx}÷{x}

Integrating both sides gives:

{dv}÷{g(v) – v} = {dx}÷{x}

Example

Consider the differential equation:

{dy}÷{dx} = {x + y}÷{x – y}

To verify if it’s homogeneous, check:

f(tx, ty) = {tx + ty}÷{tx – ty} = {x + y}÷{x – y} = f(x, y)

Thus, the equation is homogeneous. Using the substitution ( y = vx), we get:

{dy}÷{dx} = v + x{dv}÷{dx}

Substituting into the original equation:

v + x{dv}÷{dx} = {x + vx}÷{x – vx} = {1 + v}÷{1 – v}

This simplifies to:

v + x{dv}÷{dx} ={1 + v}÷{1 – v}

Rearranging, we have:

x{dv}÷{dx} = {1 + v}÷{1 – v} – v

x{dv}÷{dx} = {1 + v – v + v²}÷{1 – v}

x{dv}÷{dx} = {1 + v²}÷{1 – v}

Separating variables:

{(1 – v)}÷{1 + v²} dv = {dx}÷{x}

Integrating both sides:

{1 – v}÷{1 + v²} dv = {dx}÷{x}

{dv}÷{1 + v²} – {v dv}{1 + v²} = ln |x| + C

arctan(v) – {1}÷{2}ln |1 + v²| = ln |x| +

arctan left({y}÷{x}right) – {1}÷{2}ln left(1 + left({y}÷{x}right)²right) = ln |x| + C

This solution relates to y and x.

Conclusion

Homogeneous differential equations, through their specific structural properties, lend themselves to a systematic solution approach via substitution and transformation into separable forms. This method not only simplifies the process of solving such equations but also demonstrates the elegance and power of mathematical techniques in addressing complex problems.

Understanding and mastering these techniques are crucial for tackling a wide range of problems in differential equations. They provide valuable tools for both theoretical analysis and practical applications in various scientific and engineering fields. Turito is your to-go online preparational program that preps you well for your academic and professional exams via expert-led classes and an all-inclusive curriculum.

FAQs

What are the types of homogeneous differential equations?

Homogeneous differential equations are of two types: first-order and second-order (or higher-order) homogeneous differential equations:
First-order homogeneous differential equations: These involve first derivatives and can be recognized by their ability to rewrite the ratio of the dependent and independent variables.
Second-order (or higher-order) homogeneous differential equations: These involve higher derivatives and are characterized by the property that every term of the equation is a homogeneous function of the dependent variable and its derivatives. For linear differential equations, the homogeneous form means the absence of any independent term (non-homogeneous part)

What is the significance of the degree of homogeneity in differential equations?

The degree of homogeneity of a differential equation provides important insights into the equation’s behavior and the methods used to solve it. In the context of first-order equations, the degree of homogeneity, typically associated with the function f(x, y), indicates how the function scales with its arguments.

How do homogeneous differential equations differ from non-homogeneous differential equations?

The primary distinction between homogeneous and non-homogeneous differential equations lies in the presence or absence of an independent (non-zero) term.
Homogeneous differential equations: These equations do not have any term that is independent of the dependent variable and its derivatives. In other words, every term in the equation is dependent on the variable(s) and their derivatives. For example, in the context of linear differential equations, a homogeneous equation is one where the right-hand side is zero.
Non-homogeneous differential equations: These equations include an independent term (often called the forcing term or non-homogeneous term). This means the equation can be written in a non-zero function. The presence of this term significantly affects the solution methods and the nature of the solutions. Non-homogeneous equations often require particular solutions in addition to the common solution of the associated homogeneous equation.