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Table of contents
1.
Introduction
2.
Properties of Homogeneous differential equations
3.
Examples of Homogeneous Differential equations
4.
Solving a Homogeneous Differential Equation
5.
Frequently Asked Questions
6.
Conclusion
Last Updated: Mar 27, 2024
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# Homogeneous Differential Equations

GAZAL ARORA
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## Introduction

1. What is a homogeneous differential equation?
2. How to prove a differential equation to be homogeneous?
3. How to solve a homogeneous differential equation?

Read this article till the end to answer all the questions mentioned above.

Letâ€™s start with Question 1, what is a homogeneous differential equation.

A homogeneous differential equation is a set of variables with differentiation and functions.

A function f(x, y) is called a homogeneous function if f(Î»x, Î»y) = Î»nf(x, y), for any non zero constant Î». A differential equation that contains a homogeneous function is called a homogeneous differential equation.

A homogeneous differential equation has the general form f(x, y).dy + g(x, y).dx = 0.

## Properties of Homogeneous differential equations

• The equation for the homogeneous differential equation does not include a constant term.
• The variables x and y are not included in any special functions in the homogeneous differential equation, such as logarithmic or trigonometric functions.
• The degree of x and y in Homogeneous differential equations are the same.
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## Examples of Homogeneous Differential equations

• dy/dx = x(x - y)/y2
• dy/dx = (x + y)/(x - y)
• dy/dx = (x2 + y2)/xy
• dy/dx = (x3 + y3)/(xy2 + yx2)

## Solving a Homogeneous Differential Equation

The general solution of a homogeneous differential equation is calculated by integrating the given differential equation.

A homogeneous differential equation: dy/dx = f(x, y) is solved by separating the variable and its derivative on either side, then integrating it with respect to the variable.

Note: The equation for the homogeneous differential equation does not include a constant term. A constant term exists in the linear differential equation. The solution for a linear differential equation is possible if the constant term is removed from the equation and is transformed into a homogeneous differential equation.

General steps to solve a homogeneous differential equation includes:

1. We use the substitution y = v.x to solve a homogeneous differential equation of the type dy/dx = f(x, y). This replacement makes it simple to integrate and solve.
2. The differentiation of y = vx, with respect to x, gives dy/dx = v + x.dv/dx.

3. By substituting the value of dy/dx in the expression dy/dx = f(x, y) = g(y/x) we get:

v + x.dv/dx = g(v)  ,     where v = y/x.

4. Separating the variables v and x, we get:

5. Integrate LHS and RHS,

6. The general solution of the given homogeneous differential equation is

To obtain the general solution of the homogeneous differential equation, we replace the value of v = y/x. The inclusion of +C in the solution indicates that it is a general solution. By solving and substituting the value of +C, we can obtain the particular solution to the given homogeneous differential equation.

Letâ€™s understand with an example,

Q: Solve the below-mentioned differential equation.

Solution:

1. Try to make y/x terms.

2. Now we have a function of y/x.

3. Now use Separation of Variables:

4. Substitute v = y/x

Hence Solved.

Also, Try Solving:

1. dy / dx = y*( x - y ) / x2
2. dy / dx = ( x - y ) / (x + y)
3. dy / dx = ( 3x + 2y ) / 3y
4. xCos(y/x)*dy / dx = y*Cos(y/x) + x

## Frequently Asked Questions

1. What is a homogeneous differential equation?
A homogeneous differential equation contains a set of variables with differentiation and a function. In a homogeneous differential equation, the function f(x, y) is a homogeneous function such that f(Î»x, Î»y) = Î»nf(x, y) for any non-zero constant n.

2. Why is it called a homogeneous equation?
A linear differential equation Ly = f with some function f = 0 is called a homogeneous equation because if x is a solution of Lx = 0, then kx also solves this equation for some constant k.

3. How can we prove a differential equation to be homogenous?
To prove a differential equation to be homogenous, show that it is of the form f(Î»x, Î»y) = Î»nf(x, y) for any non zero constant Î».

## Conclusion

In this article, we learned about Homogeneous differential equations. We learned that homogeneous differential equations are of the form f(Î»x, Î»y) = Î»nf(x, y) for any non zero constant Î».

We also learned about the steps required to solve a homogeneous differential equation.

Now you are all set to solve any level of the homogeneous differential equation.

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