Homogeneous equations

Homogeneous equations are equations in which all terms are of the same degree or can be made to be of the same degree by appropriate substitutions. In the context of linear algebra and differential equations, homogeneous equations refer to those that equal zero.

Linear Homogeneous Equations: In linear algebra, a homogeneous system of equations has the general form $Ax = 0$ , where $A$ is a matrix and $x$ is a vector of variables. The solutions to this system form a vector space, and the trivial solution (where all variables are equal to zero) is always a solution.
Homogeneous Differential Equations: These can take forms like $y'' + p(x)y' + q(x)y = 0$ . The solutions to these equations also include the zero function and may involve finding characteristic equations to determine the general solution.

In both contexts, the key characteristic is that the equation or system exhibits a structure that allows for scaling, and the presence of a zero output or solution. Homogeneous equations are crucial in various fields such as physics, engineering, and applied mathematics for understanding stability, equilibrium, and system behavior.

In this article

Part 1: First order homogenous equations Part 2: First order homogeneous equations 2

Part 1: First order homogenous equations

Introduction to first order homogenous equations.

When studying first-order homogeneous equations, focus on these key points:

Definition: A first-order homogeneous equation can generally be expressed in the form $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$ .
Substitution: Use the substitution $v = \frac{y}{x}$ , which leads to $y = vx$ and helps to simplify the equation.
Differential Form: After substitution, rewrite the equation in terms of $v$ and $x$ :
$\frac{dy}{dx} = v + x\frac{dv}{dx}$
Separation of Variables: The resulting equation often allows for the separation of variables, making it easier to integrate.
Integration: Integrate both sides to find a relationship between $v$ and $x$ .
Back Substitution: Substitute back $v = \frac{y}{x}$ to express the solution in terms of $y$ and $x$ .
General Solution: The general solution will often include a constant of integration.
Special Cases: Be aware of special cases or specific forms that might require different approaches.
Graphical Interpretation: Understand how the solutions represent families of curves and their behavior in different quadrants.
Boundary Conditions: When provided, apply boundary conditions to determine particular solutions.

These points form the foundational understanding necessary for tackling first-order homogeneous equations.

Part 2: First order homogeneous equations 2

Another example of using substitution to solve a first order homogeneous differential equations.

When studying "First Order Homogeneous Equations," focus on the following key points:

Definition: Understand that a first-order homogeneous differential equation has the form $dy/dx = f(y/x)$ .
Homogeneity: Recognize that the function $f(y/x)$ is homogeneous of degree zero, meaning it has the property $f(ty/tx) = f(y/x)$ for any non-zero $t$ .
Change of Variables: Learn to make the substitution $v = y/x$ (where $y = vx$ ) to simplify the equation into a separable form.
Separation of Variables: Be able to separate variables and integrate both sides of the equation.
General Solution: Identify how to find the general solution, often involving an implicit or explicit formulation.
Particular Solutions: Study how to find particular solutions based on initial conditions.
Graphical Interpretation: Understand the geometric implications of the solutions and the behavior of the function $f$ .
Applications: Familiarize yourself with the applications of first-order homogeneous equations in modeling real-world situations.

By mastering these concepts, you'll be well-prepared to tackle first-order homogeneous equations effectively.

Homogeneous equations

Part 1: First order homogenous equations

Part 2: First order homogeneous equations 2

Related topics

Intro to differential equations

Slope fields

Euler's Method

Separable equations

Exponential models

Logistic models

Exact equations and integrating factors