Formation of a Differential Equation

The presence of unknown derivatives in the equation is called a differential equation.

These derivatives are related to other functions in the differential equation case.

The function’s derivatives are defined as the rate of change of the function at a particular point.

In real world functions are used for studying the solution that satisfies the equation and the properties of the solution.

It has been used in many fields such as physics, engineering, and biology.

WHAT ARE DIFFERENTIAL EQUATIONS?

• If the equation has one unknown derivative, it is called a differential equation. It may be either a partial equation or an ordinary equation.
• Suppose if the rate of change of function q concerning p is inversely proportional to q, we express it as

           dq/dp = k/q

• An equation involving independent variable (x) ,dependent variable (y) and different possible derivatives of dependent variables (y) with respect to independent variable (x) .
• The differential equation represents nothing but the change of something concerning something and the relationship between them.
• For example, Q=F(p)

   where,

1. Q – dependent variable.
2. p is the independent variable.
3. F is an unknown function.

Here are a few differential equation formulas for a clear understanding 

ORDER OF DIFFERENTIAL EQUATIONS

It represents the highest order of the derivative. Here are some of the examples

1. (dy/dx) = sec x
2. (d2y/dx2) + p2y = 0
3. (2d2y/dt2) + (5d2x/dt2) = 8x
4. (d3y/dx3) + 7x(dy/dx) – 9xy = 10
5. (rdr/dθ) + tanθ = 10

• In the first case, the order is 1
• In the second case, the order is 2 as the equation has second-order derivative d2 y/ dx2
• In the third case, the order is 2 as the equation has second-order derivative d2 y/ dx2
• In the fourth case, the order is 3 as the equation has second-order derivative d3 y/ dx3
• In the fifth case, the order is 1

             dy/dx  + Y = 10

where dy/dx = is differential derivatives

First-order differential equation

The equation is of the first order since only the first derivative dy/dx is involved (and not higher-order derivatives).

. The first order derivatives are represented by 

dy/dx =f(x,y)= y’

The example of a first-order differential equation is given below.

For example – (dy/dx) = tan x

Second-order differential equation

If the equation has the highest of a second-order derivative is called the second-order differential equation. It is represented by :

               ddxdydx = d2ydx2=f”(x) = y”

Here is an example of a second-order differential equation 

5d2ydx2 + 8dydx= 12x

DEGREE OF DIFFERENTIAL EQUATIONS

• The highest-order derivatives power is called the degree of the differential equation.
• For the differential equation in the polynomial form the power of the highest order term  is called the degree.
• Choose the positive integer of the find about the degree.

                     ( d2ydx2 )4  + 7 ( dydx )8  + 9y = 7 sec 7x

,where d2ydx2  is the highest order of the equation, so the highest degree of the equation is 4.

TYPES OF DIFFERENTIAL EQUATION

There are two types of differential equations. They are

1. Ordinary differential equation
2. Partial differential equation 

Formation of ordinary differential equations

• ODE – ordinary differential equation where only one variable is present. But it has lots of derivatives concerning variables.
• It has one dependent variable y and an independent variable concerning x
• For example here are some derivatives y’+y”+y”’+y””+……… + y,,,,,,…….., with respect x
• The ordinary differential equation is further classified into two. They are

Example: (d2y/dx2) + (dy/dx) = 3y tanx

1. Homogeneous differential equation
2. Nonhomogeneous differential equation 

HOMOGENEOUS DIFFERENTIAL EQUATION

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

Homogeneous differential equation

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

                dx/dy = F(x,y)

Homogeneous function

                 f (δx,δy) =  δ n f(x,y)

                    where δ  is non zero constant

• No constant term is present in the homogeneous differential equation.
• If we remove the constant term from the linear differential equation, then the equation would turn into the homogeneous differential equation.
• No variable is present in the special functions like the logarithm of trigonometry.

Here are some example 

• dy/dx = (6x + y)/(10x – y)
• dy/dx = x(5x -12y)/y2
• dy/dx = (2×2 + 5y2)/xy
• dy/dx = (3x + 9y)/(x – 2y)
• dy/dx = (11×3 + y3)/(5xy2 + 6yx2)

You can substitute x and y in all the above examples to prove the homogeneous differential equation.

• x = δx
• y = δy

Substitute x/y = v or x= vy , when the homogeneous differential equation is in the form of dx/dy=f(x,y) and has the homogeneous function f(x,y). 

NON- HOMOGENOUS DIFFERENTIAL EQUATION

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

dydx=x-y+2x+y+3

LINEAR DIFFERENTIAL EQUATION

The differential equation of the form

            (dy/dx) + Ry = S where R, S are the constant or the function of y

• The general solution is given as

  Y ( integrating factors) =   (integrating factors).dx + c

integrating factors =eR dx

Formation of partial differential equations

Partial differential equations help to relate a function containing several variables to their partial derivatives. Such equations are known as  Partial differential equation(PDE)

Here are a few examples that are

• 5𝛿u/ dx + 7𝛿u/dy = 0,
• 21𝛿2u/𝛿x2 + 8𝛿2u/𝛿y2 = 0

Formation of differential equation

The formation of the differential equation is used in real life.

• The second law of Newton was explained by the differential equation m d2x(t)dt2= f(t,x(t)), .
• It is the second differential equation where x is the unknown height as the function of time.
• The population increased over the period. If p >0 is the growth rate, then the differential equation modeling the population is given as dN/dt = pN where p is the growth rate.

• The vapor pressure V change concerning the temperature T as it is directly proportional to vapor pressure and inversely proportional to the square of temperature

           dVdt=k Vt2

Conclusion

Differential conditions assume a significant part in utilizations of sciences and engineering. It emerges in a wide assortment of engineering applications for example electromagnetic hypothesis, signal handling, computational liquid elements, and so forth These equations can be typ-ically settled utilizing either scientific or mathematical techniques.