Homogeneous Differential Equations

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.

What is a Homogeneous Function?

A function f(x, y) in x and y is said to be a homogeneous function if the degree of each term in the function is constant (say p). For example, f(x, y) = (x² + y² – xy) is a homogeneous function of degree 2 where p = 2. Similarly, g(x, y) = (x³ – 3xy² + 3x²y + y³) is a homogeneous function of degree 3 where p = 3. 

In general, a homogeneous function ƒ(x, y) of degree n is expressible as:

ƒ(x, y) = λⁿ ƒ(y/x)

Homogenous function and homogenous differential equation are represented in the image below,

What is a Homogeneous Differential Equation?

All the equations of the following form are Homogenous Differential Equations.

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

where,
f(x, y) and g(x, y) are homogeneous functions of the degree n.

In simple words, a differential equation in which all the functions are of the same degree is called a homogeneous differential equation. For example, dy/dx = (x² – y²)/xy is a homogeneous differential equation. 

Examples of Homogeneous Differential Equations

Some more examples of the homogenous differential equation are,

• dy/dx = (2x + 3y)/(7x – y)
• dy/dx = 3x(x – y)/2y²
• dy/dx = (2x³ + 2xy²)/(y³ + 3yx²)
• dy/dx = (11x² + xy)/2y²

In the above example, the degree of each term in the function is constant and hence, they are differential equations.

How to Solve Homogeneous Differential Equations?

Homogenous differential equations are equations that contain a homogenous function. We can solve a homogeneous differential equation of the form dx/dy = f(x, y) where, f(x, y) is a homogeneous function, by simply replacing x/y to v or putting y = vx. Then after solving the differential equation, we put back the value of v to get the final solution. The detailed step for solving the Homogeneous Differential Equation i.e., dy/dx = y/x.

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

Now, if y = vx

then, dy/dx = v + xdv/dx

Substituting these values in the given D.E

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

v + xdv/dx = vx/x

⇒ v + xdv/dx = v

⇒ xdv/dx = 0

⇒ dv = 0

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

Integrating both sides,

∫dv = 0

⇒ v = c

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

Substituting y/x = v

⇒ y/x = c

⇒ y = cx

This is the required solution of the given homogeneous differential equation

Learn more about, How to solve Differential Equations?

Non-Homogeneous Differential Equation

Any differential equation which is not Homogenous is called a Non-Homogenous Differential Equation. The general form of the linear non-homogeneous differential equation of second order is,

y”+a(t)y’+b(t)y = c(t)

Where,

• y” represents the second-degree differentiation of y, and 
• c(t) is a non-zero function.

The above non-homogenous differential equation can be converted to a homogeneous differential equation and the related DE is,

y”+a(t)y’+b(t)y = 0

This equation is also called the complementary equation to the given non-homogeneous differential equation.

Examples on Homogeneous Differential Equations

Example 1: Solve dy/dx = y² – x²/2xy

Solution:

Clearly, since each of the functions (y² – x²) and 2xy is a homogeneous function of degree 2, the given equation is homogeneous.

Putting y = vx and dy/dx = v + x dy/dx, the given equation becomes

⇒ v + x dv/dx = (v²x² – x²)/2vx²

⇒ v + x dv/dx = v² – 1/2v   [after dividing (v²x²/2vx² – x²/2vx²)]

⇒ x dv/dx = ((v² – 1/2v) – v)

⇒ x dv/dx = -(1 + v²)/2v

⇒ 2v/(1 + v²)dv = -1/x dx

⇒ ∫2v/(1 + v²)dv = -∫1/x dx [Integrating both the sides]

⇒ log | 1 + v² | = -log | x | + log C

⇒ log | 1 + v² | + log | x | = log C

⇒ log | x(1 + v²) | = log C

⇒ x(1 + v²) = ±C

⇒ x(1 + v²) = C₁

⇒ x(1 + y²/x²) = C₁  [Putting the original value of v = y/x]

⇒ x² + y² = xC₁, which is the required solution  

Example 2: Solve (x√(x² + y²) – y²)dx + xy dy = 0

Solution:

The given equation may be written as 

dy/dx = y² – x√(x² + y²)/xy, which is clearly homogeneous 

Putting y = vx and dy/dx = v + x dv/dx in it, we get

⇒ v + x dv/dx = {v²x² – x√(x² + v²y²)}/vx²  

⇒ x dv/dx = [{v² – √(1 + v²)}/v – v]

⇒ x dv/dx = -√(1 + v²)/v

⇒ ∫v/√(1 + v²)dv = -∫dx/xc [Integrating both the sides]

⇒ √(1 + v²) = -log | x | + C

⇒ √(x² + y²) + x log | x | = Cx, which is the required solution after putting the value of v = y/x.

Example 3: Solve x dy/dx – y = √(x² + y²)

Solution:

The given equation may be written as dy/dx = {y + √(x² + y²)}/x ,which is clearly homogeneous.

Putting y = vx and dy/dx = v + x dv/dx in it, we get

⇒ v + x dv/dx = {vx + √(x² + v²x²)}/x 

⇒ v + x dv/dx = v + √(1+v²) [After dividing the {vx + √(x² + v²x²)}/x]

⇒ x dv/dx = √(1 + v²) [v on the both sides gets cancelled]

⇒ dv/√(1+v²) = 1/x dx [after rearranging]

⇒ ∫dv/√(1+v²) = ∫1/x dx [integrating both sides]

⇒ log | v | + √(1 + v²) | = log | x | + log C

⇒ log | {v + √(1 + v²)}/x | = log | C |

⇒ {v + √(1 + v²)}/x = ±C

⇒ v + √(1 + v²) = C₁x, where C₁ = ±C

⇒ y + √(x² + y²) = C₁x², which is the required solution after putting the value of v = y/x

Example 4: Solve (x cos(y/x))(y dx + x dy) = y sin(y/x)(x dy – y dx)

Solution:

The given equation may be written as

(x cos(y/x) + y sin(y/x))y – (y sin(y/x) – x cos (y/x)) x . dy/dx = 0

⇒ dy/dx = {x cos (y/x) + y sin(y/x)}y / {y sin(y/x) – x cos(y/x)}x

⇒ dy/dx = {cos (y/x) + (y/x)sin(y/x)}(y/x) / {(y/x)sin(y/x) – cos(y/x)} [Dividing numerator and denominator by x²], which is clearly homogeneous ,being a function of (y/x).

Putting y = vx and dy/dx = (v + x dv/dx) in it, we get

⇒ v + x dv/dx = v(cos v + sin v)/(v sin v – cos v)

⇒ x dv/dx = [v(cos v + sin v)/(v sin v – cos v) -v]

⇒ x dv/dx = 2vcos v/(v sin v – cos v)

⇒ ∫{(v sin v – cos v)/2vcos v}dv = ∫x dx  [Integrating both sides]

⇒ ∫tan v dv – ∫ dv/v = ∫ 2/x dx

⇒ -log | cos v | – log | v | + log C = 2 log | x |

⇒ log | cos v | + log | v | + 2log | x | = log | C |

⇒ log | x²v cos v | = log | C |

⇒ | x²v cos v | = C [After cancelling log on the both sides]

⇒ x²v cos v = ± C

⇒ x²v cos v = C₁ [here we taking ±C = C₁]

⇒ xy cos(y/x) = C_1, which is the required solution after putting the actual value of v = y/x

FAQs on 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.