Ordinary Differential Equations
An ordinary differential equation contains the derivative of an unknown function. The ordinary differential equation is an equation having variables and a derivative of the dependent variable with reference to the independent variable. The two types of ordinary differential equations are the homogeneous differential equation and non-homogeneous differential equation.
Let us learn more about the ordinary differential equations along with the process of finding their order, degree, and solution.
What is an Ordinary Differential Equation?
An ordinary differential equation (ODE) is an equation with ordinary derivatives (and NOT the partial derivatives). A differential equation is an equation having variables and a derivative of the dependent variable with reference to the independent variable. A differential equation contains at least one derivative of an unknown function, either an ordinary derivative or a partial derivative. In particular, an ordinary differential equation has ordinary derivatives. Here the ordinary differential equations would be commonly referred to as only differential equations.
The notations used for the derivatives in these ordinary differential equations are dy/dx = y', d2y/dx2 = y'', d3y/dx3 = y''', dny/dxn = yn. A few examples of ordinary differential equations are as follows.
- (dy/dx) = sin x
- (d2y/dx2) + k2y = 0
- (d2y/dt2) + (d2x/dt2) = x
- (d3y/dx3) + x(dy/dx) - 4xy = 0
- (rdr/dθ) + cosθ = 5
Order and Degree of Ordinary Differential Equation
The two important aspects of ordinary differential equations, is the order and the degree of the differential equation. Let us look into each of it in detail.
Order of Ordinary Differential Equations
The order of a differential equation is the order of the highest derivative of the dependent variable with respect to the independent variable. Consider the following differential equations, dy/dx = ex, (d4y/dx4) + y = 0, (d3y/dx3) + x2(d2y/dx2) = 0. In these differential equations, the highest derivatives are of first, fourth and third order respectively and hence their orders are 1, 4, and 3 respectively.
First Order Differential Equation: It is the first-order differential equation that has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation: The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Degree of Ordinary Differential Equations
If a differential equation is expressible in a polynomial form, then the integral power of the highest order derivative that appears is called the degree of the differential equation. The degree of the differential equation is the power of the highest ordered derivative present in the equation. To find the degree of the differential equation, we need to have a positive integer as the index of each derivative. Example: \((\dfrac{d^4y}{dx^4})^3+ 4(\dfrac{dy}{dx}) ^7 + 6y = 5cos 3x\)
Here the order of the differential equation is 4 and the degree is 3. The order and degree of a differential equation are always positive integers. Further, if a differential equation is not expressible in terms of a polynomial equation having the highest order derivative as the leading term, then that degree of the differential equation is not defined.
Types of Ordinary Differential Equations
The ordinary differential equations are broadly classified as homogeneous differential equations and non-homogeneous differential equations. Let us check more about each of these two types of differential equations.
Homogeneous Differential Equation
A differential equation in which the degree of all the terms is the same is known as a homogeneous differential equation. In general they can be represented as P(x,y)dx + Q(x,y)dy = 0, where P(x,y) and Q(x,y) are homogeneous functions of the same degree. Some of the examples of homogeneous hifferential equations are as follows.
- y + x(dy/dx) = 0
- x4 + y4(dy/dx) = 0
Note: xy(dy/dx) + y2 + 2x = 0 is not a homogeneous differential equation
Non-Homogeneous Differential Equation
A differential equation in which the degree of all the terms is not the same is known as a nonhomogeneous differential equation. For example, xy(dy/dx) + y2 + 2x = 0 is not a homogeneous differential equation. One of the types of a non-homogeneous differential equation is the linear differential equation, which is similar to the linear equation.
Linear differential equation is an equation having a variable, a derivative of this variable, and a few other functions. The standard form of a linear differential equation is dy/dx + Py = Q, and it contains the variable y, and its derivatives. Here P and Q in this differential equation are either numeric constants or functions of x. This is referred to as a linear differential equation in y. Similarly, we can write the linear differential equation in x also. The linear differential equation in x is dx/dy + P1x = Q1.
The differential is a first-order differentiation and is called the first-order linear differential equation. Some of the examples of linear differential equation in y are dy/dx + y = cos x, dy/dx + (-2y)/x = x2.e-x and the examples of linear differential equation in x are dx/dy + x = sin y, dx/dy + x/y = ey. dx/dy + x/(ylogy) = 1/y.
Solution of Ordinary Differential Equations
For a given ordinary differential equation, y = φ(x) the solution curve (integral curve) is called the solution of the ordinary differential equation. The ordinary differential equation has infinitely many solutions. Solving an ordinary differential equation is referred to as integrating a differential equation since the process of finding the solution to a differential equation involves integration. A solution of an ordinary differential equation is an expression of the dependent variable with reference to the independent variable, which satisfies the differential equation.
General Solution: The solution which contains arbitrary constants is called the general solution. The general solution can contain numerous arbitrary constants.
Particular Solution: The solution free from arbitrary constants and is obtained by substituting values to the arbitrary constants of the general solution is called the particular solution of the differential equation.
The result of eliminating one arbitrary constant yields a first-order differential equation and that of eliminating two arbitrary constants leads to a second-order differential equation and so on. Let us understand solving the differential equation by an example.
(dy/dx) = x2y + y
Step 1: Divide the above differential equation by y. (We separate the variable)
(1/y)(dy/dx) = (x2 + 1)
We consider y and x both as variables and rewrite this as
(dy/y) = (x2 + 1)dx
Step 2: Now integrate L.H.S. with respect to y and R.H.S with respect to x.
∫(1/y)dy = ∫(x2 + 1)dx
Step 3: After integrating, we get:
log y = (x3/3) + x + c
Hence, this is the general solution of the ordinary differential equation as it contains the arbitrary constant C. Further, for different values of C, we can obtain the respective particular solutions. For more detailed information about the solutions of differential equations, click here.
☛ Related Topics:
The following topics help in a better understanding of ordinary differential equations.
FAQs on Ordinary Differential Equations
What are Ordinary Differential Equations with Examples?
The ordinary differential equations are equations which contain the ordinary derivatives such as dy/dx, d2y/dx2, etc. The ordinary differential equations are also as only differential equations. The notations used for the derivatives in these ordinary differential equations are dy/dx = y', d2y/dx2 = y'', d3y/dx3 = y''', dny/dxn = yn. A few examples of ordinary differential equations are as follows.
- (dy/dx) = sin x
- (d2y/dx2) + k2y = 0
- (d2y/dt2) + (d2x/dt2) = x
- (d3y/dx3) + x(dy/dx) - 4xy = 0
How Do You Find Ordinary Differential Equation?
The ordinary differential equation can be identified if there is a derivative expression of the dependent variable with reference to only one independent variable. If the derivative expression involves the derivative of the dependent variable with reference to more than one independent variable, then it is called a non-homogeneous differential equation.
How Many Types of Ordinary Differential Equations Are There?
The two types of ordinary differential equations are homogeneous differential equations and non-homogeneous differential equations.
- In a homogeneous differential equation, the degree of all the terms is the same. In general they can be represented as 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 differential equation in which the degree of all the terms is not the same is known as a non-homogeneous differential equation. For example, xy(dy/dx) + y2 + 2x = 0 is not a homogeneous differential equation.
What Are the Applications of Ordinary Differential Equations?
Ordinary differential equations applications in real life include its use to calculate the movement or flow of electricity, to study the to and fro motion of a pendulum, to check the growth of diseases in graphical representation, mathematical models involving population growth, and in radioactive decay studies.
What is the Difference Between Ordinary Differential Equations and Partial Differential Equation?
A differential equation contains at least one derivative of an unknown function, either an ordinary derivative or a partial derivative. The differential equations involving the derivative of one dependent variable with reference to more than one independent variable is called a partial differential equation and the differential equation involving the derivative of one dependent variable with reference to another independent variable is called the ordinary differential equation.