Homogeneous Differential Equations

By definition, a homogenous equation must have zero on the right side of the = sign. It varies from non-homogeneous equations as the latter has a function of the independent variable on the right-hand side of the equation. In calculus, homogeneous differential equations are simply the differential equations with a homogenous function. The homogeneous function is a function of multiple variables such that, if all the arguments are multiplied by a scalar, then the value of the function is multiplied by the power of that scalar. If a function f(x,y) is expressed by x = cx and y = cy to generate a new function f(cx, cy) = cⁿ(x,y), such that the constant can be assumed as the n^th power of the exponent, then the equation is homogeneous. An equation presenting the form of y(n) p(x). y^{(n)}. P(x) y(n)p(x) is termed as a homogeneous linear equation. Linear signifies that a derivative of y times x’s function while homogenous indicates 0 on the right side. A homogeneous equation always has a consistent solution because of zero. Some common homogeneous differential equation examples are provided below:

dy/dx = (x+y)/(x-y)

dy/dx = (x³ + y³)/(xy² + yx²)

dy/dx = (3x + y)/(x –y)

Homogeneous Differential Equations

Johann Bernoulli proposed homogeneous differential equations in integration. Two ways by which a differential equation can be homogenous are:

1. First-order equations are considered to be homogenous if, f(x,y)dy = m(x,y)dx

Here, f and m are homogenous functions of x and y. They have the same degree.

Similarly, dx/x = q (µ) dµ; as we change the variable y =µx. This will help to perform the integration.

1. Secondly, a differential equation may be homogenous if it has the homogenous function of the present unknown functions and their derivatives.

According to Bernoulli, a first order differential equation M(x,y)dx + N(x,y)dy = 0 is homogenous if both the functions have equal degree n.

M(γx, γy) = γⁿM(x,y) = N(γx, γy)

Therefore, M(γx, γy)/N(γx, γy) = M (x,y)/N(x,y).

The problems can be solved using the method of separation of variables. Before that, we need to introduce a new variable.

We will see this in a simple example.

Problem:

dy/dx = x² +y² /xy

Solution:

We need to separate the terms on the right side, so:

x² +y² /xy

= x² /xy + y² /xy

=x/y + y/x

Using reciprocal method we can write:

(y/x)^-1 + (y/x)

HERE WE GO, we have a function of (y/x).

Returning to the sum,

dy/dx = (y/x)^-1 + (y/x)

Let us consider y = vx.

So, dy/dx = v + x.dv/dx: v + x.dv/dx = v^-1 + v

x.dv/dx = v^-1 [subtracting v from both ends]

In this stage we are going to implement separation of variables.

v dv = 1/x. dx

Integrating both ends, ∫v.dv = ∫1/x.dx

v²/2 = ln(x) + C

Let’s consider C = ln(k)

Therefore, v²/2 = ln(x) + ln(k)

Or, v²/2 = ln(kx)

Simplification will lead to:

v = ± √(2 ln(kx))

After putting back y/x, y/x = ± √(2 ln(kx))

Or, y = ± x√(2 ln(kx))

We have solved the equation.

Homogeneous Linear Equation

A system of the equation can be termed as homogeneous and linear given, the constant term is found to be zero.

C₁₁x₁ + C₁₂x₂ + … + C_1nx_n = 0 … 1

C₂₁x₁ + C₂₂x₂ + … + C_2nx_n = 0 … 2

C_m1x₁ + C_m2x₂ + … + C_mnx_n = 0 … 3

A homogeneous equation is bound to have one obvious solution. Here, x₁ = x₂ = x_n = 0 for all the above three equations.

To have infinite solutions, the value of n must be greater than m (n > m).

Also, we can conclude that there are (n-m) free variables that can uptake any value.

Problem

Solve 2y’’ – 5y’ – 3y = 0

Solution

Let us transform the equation into algebraic form by replacing y’’ with m², y’ with m, and y with 1.

New equation: 2m² – 5m – 3 = 0

The roots of this equation are – ½ and 3. This confirms that e^-x/2 and e^3x are two linearly independent solutions.

Finally, the general solution is presented as: y = C₁ exp (-x/2) + C₂ exp (3x)

Conclusion

In this article, we discuss the various ideas of solving homogeneous equations. We know that nonhomogeneous differential equations have terms involving x on the right side. This characteristic helps us to identify whether a DE is homogeneous or nonhomogeneous. Homogeneous linear equations with constant coefficients are given as: y’’ + by’ + cy = 0. The differential equation plays a role in modifying the output after we start with independent variables for specified inputs. The roots of homogeneous differential equations determine the general solution of the equation.