Formation of Differential Equations

A differential equation can be defined as an equation that consists of a function {say, F(x)} along with one or more derivatives { say, dy/dx}. The functions of a differential equation usually represent the physical quantities whereas the rate of change of the physical quantities is expressed by its derivatives. A differential equation is actually a relationship between the function and its derivatives.

For Example - if we consider y as a function of x then an equation that involves the derivatives of y with respect to x (or the differentials of y and x) with or without variables x and y are known as a differential equation. Given below are some examples of the differential equation:

1. x \[\frac{dy}{dx}\] = 2y

2. xdy + \[y^{2}\] dx = dx

3. \[\frac{d^{2}y}{dx^{2}}\] = \[\frac{dy}{dx}\]

4. \[y^{2}\]  \[\left ( \frac{dy}{dx} \right )^{2}\] - x \[\frac{dy}{dx}\] = \[x^{2}\]

5. \[\left ( \frac{d^{2}y}{dx^{2}} \right )^{2}\] = x \[\left (\frac{dy}{dx} \right )^{3}\]

6. \[x^{2}\] \[\frac{d^{3}y}{dx^{3}}\] - 2y \[\frac{dy}{dx}\] = x

7. \[\left \{ 1 + \left ( \frac{dy}{dx} \right )^{2} \right \}^{\frac{3}{2}}\] = a \[\frac{d^{2}y}{dx^{2}}\]  or,  \[\left \{ 1 + \left ( \frac{dy}{dx} \right )^{2} \right \}^{3}\] = \[a^{2}\] \[\left (\frac{d^{2}y}{dx^{2}}  \right )^{2}\]

Order and Degree of a Differential Equation

A differential equation is an equation that has one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable).

\[\frac{dy}{dx}\] = f(x)

In this case, “x” is an independent variable and “y” is a dependent variable

For example, \[\frac{dy}{dx}\] = 7x

A differential equation consists of derivatives which are either partial derivatives of ordinary derivatives. The derivative is used to represent a rate of change, and the differential equation draws a relationship between the quantity that is continuously varying to the change in another given quantity. There are some differential equation formulas to find the solution of the derivatives. In differential equations, order and degree are the main parameters for classifying different types of differential equations. The order of differential equations is actually the order of the highest derivatives (or differential) in the equation. 

First-order differential equation: First-order differential equation having degree equal to 1. All linear equations in the form of derivatives are in the first order.  It has only the first derivative such as \[\frac{dy}{dx}\], where x and y are the two variables and is given as:

\[\frac{dy}{dx}\] = f(x, y) = y’

Thus, in the examples given above, Equations (1), (2) and (4) are of the 1st order as the equations involve only first-order derivatives (or differentials) and their powers; 

Second-order differential equation: The equation which contains the second-order derivative is called the second-order differential equation.  It is given as;

\[\frac{d}{dx}\](\[\frac{dy}{dx}\]) = \[\frac{d^{2}y}{dx^{2}}\]  = f”(x) = y”

Equations (3), (5), and (7) are of 2nd order as the highest order derivatives occurring in the equations being of the 2nd order, and equation (6) is the 3rd order. The power of the highest order derivative is called the Degree of the Differential Equation, where the original equation is given in the form of a polynomial equation in derivatives such as y’,y”, y”’, and so on.

Suppose (\[\frac{d^{2}y}{dx^{2}}\])+ 3 (\[\frac{dy}{dx}\])+y = 0 is a differential equation, then the degree of this equation is 1. The degree of a differential equation is the highest power (or degree) of the derivative of the highest order of differential equations in an equation. After the equation is cleared of radicals or fractional powers in its derivatives. In the above examples, equations (1), (2), (3) and (6) are of the 1st degree and (4), (5) and (7) are of the 2nd degree.

Thus, by definition,

• Equations (1) and (2) are of the 1st order and 1st degree;

• Equation (3) is of the 2nd order and 1st  degree;

• Equation (4) is of the 1st order and 2nd degree;

• Equations (5) and (7) are of the 2nd order and 2nd degree;

• And equation (6) is of 3rd order and 1st degree.

Formation of Differential Equations 

For every given differential equation, the solution will be of the form f ( x, y, c1, c2, …….,cn) = 0 where x and y will be the variables and c1, c2 ……. cn will be the arbitrary constants. To understand the formation of differential equations in a better way, there are a few suitable differential equations examples that are given below along with important steps. The formulas of differential equations are important as they help in solving the problems easily.

To achieve the differential equation from this equation we have to follow the following steps:

Step 1: we have to differentiate the given function w.r.t to the independent variable that is present in the equation.

Step 2: Secondly, we have to keep differentiating times in such a way that (n+1 ) equations can be obtained.

Step 3: With the help of (n+1) equations obtained, we have to eliminate the constants   ( c1, c2 … …. cn).

Solved Examples

Example 1: Find the order of the differential equation. 

3y2(\[\frac{dy}{dx}\])3 - ( \[\frac{d^{2}y}{dx^{2}}\] ) = sin(x2) 

Solution: 

The highest order derivative associated with this particular differential equation, is already in the reduced form, is of 2nd order and its corresponding power is 1. Therefore, the order of the differential equation is 2 and its degree is 1.

Example 2: Find the differential equation of the family of circles x2 + y2 = 2ax, where a is a parameter.

Solution: 

Given, x2 + y2 = 2ax ………(1) By differentiating both the sides of (1) with respect to x, we get, 

2x + 2y \[\frac{dy}{dx}\] = 2a……………(ii)

Now, eliminating a from (i) and (ii) we get, 

x2 + y2 = x (2x + 2y \[\frac{dy}{dx}\]) or, 2xy\[\frac{dy}{dx}\]  = y2  - x2

Which is the required differential equation of the family of circles (1).

Again, assume that the independent variable x, the dependent variable y, and the parameters (or, arbitrary constants) c1 and c2 are connected by the relation 

f(x, y, c1, c2) = 0 ………. (i)

Differentiating (i) two times successively with respect to x, we get, 

\[\frac{d}{dx}\] f(x, y, \[c_{1}\], \[c_{2}\]) = 0………(ii) and \[\frac{d^{2}}{dx^{2}}\] f(x, y, \[c_{1}\], \[c_{2}\]) = 0 …………(iii)

The differential equation of (i) is obtained by eliminating c1 and c2 from (i), (ii) and (iii); evidently, it is a second-order differential equation and in general, involves x, y, \[\frac{dy}{dx}\] and \[\frac{d^{2}y}{dx^{2}}\].

In general, the differential equation of a given equation involving n parameters can be obtained by differentiating the equation successively n times and then eliminating the n parameters from the (n+1) equations. 

It is widely used in fields such as physics, engineering, biology and so on. The main purpose of differential equations is the study of solutions that satisfy the equations and the properties of the solutions. 

We can easily solve differential equations by using explicit formulas. In this article, we have discussed the definition, order and degree of the differential equation with relevant examples.