Homogeneous Differential Equations

Q1: Define Homogeneous Differential Equations.

Answer:

A differential equation which is formed by the differentiation operator, function f(x, y), the dependent and independent variable is called the Homogeneous Differential Equations. We represent Homogeneous Differential Equations as,

dy/dx = f(x, y)/g(x, y)

where f(x, y) and g(x, y) are homogenous functions

Q2: What are examples of Homogeneous Differential Equations?

Answer:

Various examples of Homogeneous Differential Equations are,

• dy/dx = (2x² + 3xy)/(7xy – y²)
• dy/dx = 13x²(x – y)/12xy²
• dy/dx = (2yx³ + 2x²y²)/(xy³ + 3y²x²), etc.

Q3: What is the difference between a Homogeneous and a Non-Homogeneous Differential Equation?

Answer:

Any differential equation in which the function used is homogenous is the homogenous differential equation. Example dy/dx = x/y is a homogenous differential equation. Whereas, any differential equation other than a homogenous differential equation is a non-homogenous differential equation. Example dy/dx = sinx is a non-homogenous differential equation. 

Q4: What are the Steps to Solve a Homogeneous Differential Equation?

Answer:

Homogeneous differential equation can be easily solved using the steps discussed below,

Step 1: Put y = vx in the given differential equation.

Step 2: Simplify and then separate the independent variable and the differentiation variable on either side of the equal to.

Step 3: Integrate the differential equation so obtained and find the general solution in v and x.

Step 4: Put back the value of v to get the final solution in x and y.

Homogeneous Differential Equations are differential equations with homogenous functions. They are equations containing a differentiation operator, a function, and a set of variables. The general form of the homogeneous differential equation is f(x, y).dy + g(x, y).dx = 0, where f(x, y) and h(x, y) is a homogenous function. Homogenous functions are defined as functions in which the total power of all the terms of the function is constant. Before continuing with Homogeneous Differential Equations we should learn Homogeneous Functions first. In this article, we will learn about, Homongenous Functions, Homogeneous Differential Equations, their solutions, and others in detail.