Homogeneous Differential Equation

Differential functions are those equations that include a function’s derivative. 

Mathematically, we call a differential equation an equation involving the derivatives of one or more independent variables and the dependent variable itself. Differential equations also contain derivatives of various orders and degrees.

We can further classify the ordinary differential equation into two as:

1. homogeneous differential equation
2. Non-homogenous differential equation

Homogeneous Differential Equation

homogenous is a term used to describe first-order differential equations where we can write the equations in the following way:

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

Here, homogenous functions f and g, which have the same degree of x and y, are shown. When y = ux  is substituted, the resulting equation is as follows:

dx/x = h(u) (du)

We can easily solve this equation by integrating the two members.

It is only homogenous if the unknown function and its derivatives form a homogenous set of variables in a differential equation. There are no constant terms in linear differential equations due to this. 

When we remove the constant term from a homogeneous equation, we can deduce the solution of a linear ordinary differential equation of any order by integration.

Homogeneous Differential Equation Examples

Here are some 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)

Homogeneous Differential Equation Solved Examples

We provided some of the easy steps to solve homogeneous differential equations.

Here is the given dy/dx=Fx,y=p(x/y)

STEP 1: 

For y, use y = vx in the given equation

STEP 2: 

Differentiate the y = vx we get dy/dx=v+x *dv/dx substitute the value in the equation

We get v+x*dv/dx=p(v)

x*dv/dx=p)v)-v.

STEP 3:

If we separate the variable from the above equation we get

dv/p(v)-v = dx/x

STEP 4:

Integrate on both sides of the equation we get

∫dv/p(v)-v* dv =∫ dx/x + c

STEP 5:

After completion of integration, replace v= y/x

Non-homogenous Differential Equations

Differential equations that are not homogenous are called non-homogenous differential equations.

Second-order linear non-homogenous differential equations are represented by the following notation:

y”+p(t)y’+q(t)y= g(t)

A non-zero function g(t) is used in this case.

The corresponding homogenous equation is as follows:

y”+ p(t)y’+q(t)y = 0 in this case which is also referred to as the complementary equation.

Non-Homogenous Differential Equation Examples

Here are some examples of a non-homogenous differential equation.

• d²y/dx² − 9 y = −6 cos 3 x , 
• d²y/dx² − 9 dy/dx = −6 cos 3 x.
• d³y/dx³ + 2 dy/dx + x = 4 e – x.
• d²y/dx² – 2 dy/dx+ 5 y = 10 xy − 3 x − 3.
• d4y/dx4  − 3 dy/dx = −12 x

Non-homogenous Differential Equations Solved Examples

It is much similar to that of a linear equation, and the order of the differential equation is not identical.

For example, the differential equation of the form (dy/dx) + py = q where p, q are the constant or the function of y.

The general solution is as follows:

   y * ( integrating factors) =∫q * (integrating factors).dx + c

where integrating factors = e∫pdx

Here are some examples of non- homogeneous differential equations

d5y /dx5 + x*d³y/dx³ + y² = 7x + 5

We can give the general form of non-homogenous equation as:

Y’’ + p(x)y’ + q(x) y = g(x)

The general solution is:

y(x)= c1y1 (x) + c2 y2 (x)+ yr(x)

Conclusion

The equation that contains differentiation, set of variables, and functions of (x, y) is called a homogeneous differential equation.

The homogeneous function in the homogeneous differential equation is f (x, y), if 

f (δx,δy) =   f(x,y) 

Where δ is non-zero constant

The general form of the homogeneous differential equation is as below

f(x,y).dy + g(x,y).dx = 0

The homogeneous differential equation should have the same power for the given variables (x, y).