Formation of Differential Equations

What Is Differential Equation?

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

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. 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; 

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 degree of a differential equation is basically 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. In order 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.

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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(dy/dx)3 - d2y/dx2=sin(x/2) 

Solution 1:

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 \[x^{2}\] +  \[y^{2}\] =2ax, where a is a parameter.

Solution 2: Given, \[x^{2}\] +  \[y^{2}\] =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, 

\[x^{2}\] +  \[y^{2}\] = x \[\left ( 2x + 2y\frac{dy}{dx} \right )\] or, 2xy\[\frac{dy}{dx}\] = \[y^{2}\] - \[x^{2}\]

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) \[c_{1}\] and \[c_{2}\] are connected by the relation 

f(x, y, \[c_{1}\], \[c_{2}\]) = 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 of \[c_{1}\] and \[c_{2}\]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. 

FAQ (Frequently Asked Questions)

1. What are the conditions to be satisfied so that an equation will be a differential equation?

A differential equation must satisfy the following conditions-

• The derivatives in the equation have to be free from both the negative and the positive fractional powers if any. 

• There must not be any involvement of the derivatives in any fraction.

• There must be no involvement of the highest order derivative either as a transcendental, or exponential, or trigonometric function.

• The coefficient of every term in the differential equation that contains the highest order derivative must only be a function of p, q, or some lower-order derivative.

2. What differential equation means?

Many important problems in fields like Physical Science, Engineering, and, Social Science lead to equations comprising  derivatives or differentials when they are represented in mathematical terms. So equations like these are called differential equations. 

Therefore, an equation that involves a derivative or differentials with or without the independent and dependent variable is referred to as a differential equation. The order of a differential equation is always the order of the highest order derivative or differential appearing in the equation. one the other hand, the degree of a differential equation is the degree of the highest order derivative or differential when the derivatives are free from radicals and negative indices.