Differential Equations-Formation of a DE

Before we understand the formation of a DE, we must first understand what differential equations are. Differential equations are the combination of a type of function (f(x)) and a derivative of an unknown function (dy or dx). In the equation, the function of the differential equation (DE) represents physical quantities; on the other hand, the derivatives represent the change rate of those physical quantities.

Differential equations can also be expressed as the relationship between a function (fx) and derivatives (dy/ dx). Formation of DE is important as differential equations today are used in various fields such as biology, physics, engineering, etc., and thus can help in addressing and solving various mathematical problems. 

Differential equation definition

Now, let us understand the definition of differential equations. Differential equations can be defined as an equation consisting of an unknown derivative or a function which can either be an ordinary derivative or a partial derivative and an independent variable.

The differential equation represents the relationship between the changing quantities concerning the change in another quantity. y=f(x) is considered the function in Differential equations in which y is a dependent variable, f is an unknown function, x is an independent variable. Now that we are clear with the definition of differential equations, let us look into the formation of a DE. 

Differential equation order

The formation of a DE requires an understanding of differential equation order. Order can be the arrangement pattern that is generally followed in equations. The order of differential equations is the highest order of the present derivatives given in the equation. For example, if the differential equation is as follows,

(d5y/dx5) + y = 0

In this case, the highest order is variable to the power 5, and thus 5 can be said to be the order of differential equations. Furthermore, there are two types of orders of DE. The first one is the differential equation of first order. A first-order DE consists of all the differential equations with a degree that equals 1. When written as derivatives, every linear equation is the first -differential equation. It is represented as dy/dx = f(x, y) = y’ as it has x and y as its two variables. The second type is the differential equation of second-order that is expressed as d2y/dx2 = f” (x) = y” and it includes a second-order derivative. 

Forms of differential equations 

The main types of classifications under the formation of a DE/ forms of differential equations are –

• Ordinary Differential Equations
• Partial Differential Equations
• Homogeneous differential equations
• Non-homogeneous differential equations

Differential equations are an ODE (Ordinary Differential Equations) if they consist of only one independent variable and one or more of its derivatives concerning the variable. ODE is defined as the equations having one independent variable x, the real dependent variable y with some of its derivatives concerning x.

This equation can be either homogeneous or non-homogeneous. For example, (d2y/dx2) + (dy/dx) = 3y cos x can be called an Ordinary differential equation as it does not include any partial derivatives. Homogeneous differential equations are one in which the degree of all the terms is the same. The general representation is P(x,y)dx + Q(x,y)dy = 0, where P(x,y) and Q(x,y) are homogeneous functions of the same degree. A non-homogeneous differential equation is one in which the degree of all the terms is not the same. For example, xy(dy/dx) + y2 + 2x = 0 can be a non-homogenous DE. Partial DE includes only partial derivatives of one or more functions of two or more independent variables. 

Differential equations formation 

The derivation and DE examples can understand the Formation of a DE. For every differential equation, the solution can be expressed as f ( x, y, c1, c2, ……., cn) = 0, where y and x are the variables and c1, c2, ……., cn is the constants (arbitrary). The following steps will help in understanding the formation of a DE using examples –

• Step 1: The function there needs to be differentiated concerning the variable present in the formula/equation.
• Step 2: The equation needs to be differentiated so that the result of the (n+1) value can be obtained.
• Step 3: Now, once we have the value of (n+1), we are supposed to find the value of the constants ( c1, c2 … …. cn).

By following the above-given steps, differential equations can be obtained. Thus, these steps are called the formation of differential equations. The differential equations are solved by dividing the equation by x or y to separate the variable. Now, we need to integrate LHS concerning y and x. After integrating the LHS and RHS, we get the final differential equation solution. 

Conclusion

Formation of a DE helps understand and solve real-life issues associated with the calculation of movement or flow of electricity, motion of an object to and fro like a pendulum or even in the explanation of thermodynamics, etc. It is also used for graphical representation of medical results and to explain mathematical models based on population growth or radioactive decay.