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Homogeneous Differential Equation

A homogeneous differential equation is an equation containing a differentiation and a function, with a set of variables. The function f(x, y) in a homogeneous differential equation is a homogeneous function such that f(λx, λy) = λⁿf(x, y), for any non zero constant λ. The general form of a homogeneous differential equation is f(x, y).dy + g(x, y).dx = 0.

Let us learn more about the homogeneous differential equation, the method to solve a homogeneous differential equation, examples, faqs.

1.	What Is A Homogeneous Differential Equation?
2.	How to Solve a Homogeneous Differential Equation?
3.	Examples on Homogeneous Differential Equation
4.	Practice Questions on Homogeneous Differential Equations
5.	FAQs on Homogeneous Differential Equation

What Is A Homogeneous Differential Equation?

A differential equation containing a homogeneous function is called a homogeneous differential equation. The function f(x, y) is called a homogeneous function if f(λx, λy) = λⁿf(x, y), for any non zero constant λ. The general form of the homogeneous differential equation is of the form f(x, y).dy + g(x, y).dx = 0. The homogeneous differential equation has the same degree for the variables x, y within the equation.

Homogenous Differential Equation

The homogeneous differential equation does not have a constant term within the equation. The linear differential equation has a constant term. The solution of a linear differential equation is possible if we are able to remove the constant term from the linear differential equation and transform it into a homogeneous differential equation. Also, the homogeneous differential equation does not have the variables x, y within any special functions such as logarithmic, or trigonometric functions.

Examples of Homogeneous Differential equations.

dy/dx = (x + y)/(x - y)
dy/dx = x(x - y)/y²
dy/dx = (x² + y²)/xy
dy/dx = (3x + y)/(x - y)
dy/dx = (x³ + y³)/(xy² + yx²)

In these above examples, we can substitute x = λx, and y = λy, to prove it for the homogeneous differential equation. Also, if the homogeneous differential equation is of the form dx/dy = f(x, y), and f(x, y) is a homogeneous function, then we substitute x/y = v, or x = vy. This on further integration, and substituting back the variables x, y, gives the general solution of the homogeneous differential equation.

How To Solve a Homogeneous Differential Equation?

The general solution of the homogeneous differential equation can be obtained by the integration of the given differential equation. A homogeneous differential equation of the form dy/dx = f(x, y), is solved by first separating the variable and the derivative of the particular variable on either side and then integrating it with respect to the variable.

To solve a homogeneous differential equation of the form dy/dx = f(x, y), we make the substitution y = v.x. Here it is easy to integrate and solve with this substitution. Further the differentiation of y = vx, with respect to x we get dy/dx = v + x.dv/dx. We can substitute the value of dy/dx in the expression dy/dx = f(x, y) = g(y/x) to get the below expression.

v + x.dv/dx = g(v)

xdv/dx = g(v) - v

Separating the variables x and v, we have:

\(\dfrac{dv}{g(v) - v} = \dfrac{dx}{x}\)

Here we integrate it on both sides, which results in the following expression.

\(\int \dfrac{1}{g(v) - v}.dv = \int \dfrac{1}{x}.dx\)

The above expression gives the following solution, which is the general solution of the differential equation.

\(\int \dfrac{1}{g(v) - v}.dv = Logx + C\)

Here we substitute back the value of v = y/x, to obtain the general solution of the homogeneous differential equation. The presence of +C in the solution, refers it as a general solution, and further solving and substituting the value of +C, we can obtain the particular solution of the given homogeneous differential equation.

Related Topics

The following topics help in a better understanding of the homogeneous differential equations.

Examples on Homogeneous Differential Equation

Example 1: Show that the differential equation (x - y).dy/dx = (x + 2y) is a homogeneous differential equation.

Solution:

(x - y).dy/dx = (x + 2y) is the given differential equation.

To prove that the above differential equation is a homogeneous differential equation, let us substitute x = λx, and y = λy.

Here we have F(x, y) = \(\dfrac{(x + 2y)}{(x - y)}\)

F(λx, λy) = \(\dfrac{(λx + 2λy)}{(λx - λy)}\)

F(λx, λy) = \(\dfrac{λ(x + 2y)}{λ(x - y)}\) = λ⁰f(x, y)

Therefore, the given differential equation is a homogeneous differential equation.
Example 2: Find the solution of the homogeneous differential equation xCos(y/x).dy/dx = yCos(y/x) + x.

Solution:

The given differential equation is xCos(y/x).dy/dx = yCos(y/x) + x

dy/dx = \(\dfrac{yCos(y/x) + x}{xCos(y/x)}\)

dy/dx = \(\dfrac{x((y/x).Cos(y/x) + 1)}{xCos(y/x)}\)

dy/dx = \(\dfrac{((y/x).Cos(y/x) + 1)}{Cos(y/x)}\)

Here let us substitute y/x =v in the above expression.

dy/dx = \(\dfrac{vCosv + 1}{Cosv}\)

Here we write y/x = v as y = vx. Differentiating y = vx on both sides we obtain dy/dx = v + x.dv/dx, which is substituted in the above expression.

v + x.dv/dx = \(\dfrac{vCosv + 1}{Cosv}\)

x.dv/dx = \(\dfrac{vCosv + 1}{Cosv}\) - v

Here we separate the variables on either side of the equal to symbol.

x.dv/dx = 1/Cosv

Cosv.dv = dx/x

Integrating this expression on both sides, we have the below expression.

\(\int Cosv.dv = \int \dfrac{1}{x}.dx\)

Sinv = Logx + C

Here we substitute back y/x = v.

Sin y/x = Logx + C

Therefore, the solution of the homogeneous differential equation is Sin y/x = Logx + C.

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Practice Questions on Homogeneous Differential Equations

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FAQs on Homogeneous Differential Equation

What Is Homogeneous Differential Equation?

The homogeneous differential equation containing is a differential equation containing a homogeneous function. The homogeneous differential equation of the form dy/dx = f(x, y), has a homogeneous function f(x, y) such that f(λx, λy) = λⁿf(x, y), for any non zero constant λ. The general form of the homogeneous differential equation is of the form f(x, y).dy + g(x, y).dx = 0.

What Is the Difference Between Homogeneous and Non-Homogeneous Differential Equation?

The homogeneous differential equation consists of a homogeneous function f(x, y), such that f(λx, λy) = λⁿf(x, y), for any non zero constant λ. A non-homogeneous differential equation does not contain a homogeneous function. The example of a non homogeneous differential equation is a linear differential equation of the form dy/dx + Py = Q.

What Are the Examples of Homogeneous Differential Equation?

A few of the examples of homogeneous differential equations are as follows.

dy/dx = (x + y)/(x - y)
dy/dx = (x² + y²)/xy
dy/dx = (3x + y)/(x - y)
dy/dx = (x³ + y³)/(xy² + yx²)

What Is the Formula for Homogeneous Differential Equation?

The general form of a homogeneous differential equation is f(x, y).dy + g(x, y).dx = 0. Here the function f(x, y) is a homogeneous function such that f(λx, λy) = λⁿf(x, y), for any non zero constant λ. Also, there is no defined formula for a homogeneous differential equation, or to find the solution of it. The homogeneous differential equation is solved through a sequence of steps.

What Are the Steps to Solve Homogeneous Differential Equation?

The homogeneous differential equation of the form dy/dx = f(x, y), can be solved through the following sequence of steps.

Step - 1: Substitute y = vx in the given differential equation.
Step - 2: Separate the variables and the differentiation of the variables on either side of the equals to symbol.
Step - 3: Find the integration of the variables and to find the general solution containing v and x.
Step - 4: Substitute back the value of v to get the general solution in the variables x and y.

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