Homogeneous differential equations

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.

Homogeneous is a term used to describe first-order differential equations where the equations can be written in the following way:

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

Here, homogeneous 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)

which can be easily solved by integrating the two members.

It is only homogeneous if the unknown function and its derivatives form a homogeneous set of variables in a differential equation. There are no constant terms in linear differential equations as a result of this. When the constant term is removed from a homogeneous equation, the solution of any linear ordinary differential equation of any order can be deduced by integration.

How to solve homogeneous differential equations

An integrated homogeneous differential equation can be found out when the differential equation is solved. To solve a homogeneous differential equation, one must first separate the variable and its derivative on either side of the equation and then integrate this integral with respect to the variable.

Homogeneous differential equations of the form dy/dx = f can be solved using the substitution y = vx. Using this method simplifies the process of integrating and resolving the problem.

 dy/dx = v + x.dv/dx, is obtained as we continue to differentiate y = vx.

 dy/dx subtracted from the previous expression yields the following result.

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

x(dv/dx) = g(v) – v

When x and v are separated, we get:

dv/(g(v)−v)=dx/x

The following expression is the result of integrating it on both sides.

∫1/(g(v)−v)dv=∫1/x.dx

The general solution to the differential equation is given by the above expression.

The differential equation

∫1/(g(v)−v ).dv=log x+C

We can now solve the homogeneous differential equation for the general solution by replacing v = y/x. 

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)

Non-homogeneous differential equations

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

Second-order linear non-homogeneous 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 homogeneous equation is as follows:

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

Conclusion

Mathematicians have long been familiar with the formula Mdx + Ndy = 0, which can be used to express a first-order differential equation and the first-degree differential equation. Furthermore, the equation is said to be homogeneous if M and N are both homogeneous functions of equal degree in the two variables.

Many important problems in physics, engineering, and social science can be expressed mathematically as differential or derivative equations. The term ‘differential equation’ is used to describe this type of equation. If the independent and dependent variables aren’t present, then the equation isn’t considered to be a differential equation. It is important to distinguish between order and degree in differential equations. The highest order derivative or differential that does not contain radicals or negative indexes determines the degree of a differential equation.