Order and Degree of Differential Equation
Order and degree of a differential equation is helpful to solve the differential equation. The differential equations can be comparable with the polynomial expressions, and the order and degree of the differential equation helps in knowing the steps required to solve the differential equation and the number of possible solutions of the differential equation.
Let us learn more about how to find the order and degree of the differential equation, with examples and FAQs.
How to Find Order and Degree of Differential Equation?
The order and degree of a differential equation help us to identify the type and complexity of a differential equation. Similar to a polynomial equation a differential equation has a differential of the dependent variable with reference to the independent variable, and here the order and degree of the differential equation are helpful to find the solutions of the differential equation.
The order of the differential equation can be found by first identifying the derivatives in the given expression of the differential equation. The different derivatives in a differential equation are as follows.
 First Derivative:dy/dx or y'
 Second Derivative: d^{2}y/dx^{2}, or y''
 Third Derivative: d^{3}y/dx^{3}, or y'''
 nth derivative: d^{n}y/dx^{n}, or y^{''''.....n times}
Further, the highest derivative present in the differential equation defines the order of the differential equation, and the exponent of the highest derivative represents the degree of the differential equation. Similar to a polynomial equation in variable x, a differential equation has derivatives of the dependent variable with respect to derivatives of the independent variable.
Let us now understand more about each of the order of differential equations and the degree of the differential equation.
Order of Differential Equation
The order of a differential equation is the highest order of the derivative appearing in the differential equation. Consider the following differential equations,
dy/dx = e^{x}, (d^{4}y/dx^{4}) + y = 0, (d^{3}y/dx^{3})^{2} + x^{2}(d^{2}y/dx^{2}) + xdy/dx + 3= 0
In above differential equation examples, the highest derivative are of first, fourth and third order respectively.
First Order Differential Equation
You can see in the first example, it is a firstorder 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’
SecondOrder Differential Equation
The equation which includes secondorder derivative is the secondorder differential equation. It is represented as; d/dx(dy/dx) = d^{2}y/dx^{2} = f”(x) = y”
Degree of Differential Equation
The degree of a differential equation is the highest power of the highest order derivative in a differential equation. The degree of the differential equation is always a positive integer. The order of the differential equation is to be identified first, and then the degree of the differential equation can be identified. The degree of the differential equation is comparable to the degree of the variable in a polynomial expression. A few examples of differential equations are as follows.
 7(d^{4}y/dx^{4})^{3} + 5(d^{2}y/dx^{2})^{4}+ 9(dy/dx)^{8} + 11 = 0
 (dy/dx)^{2} + (dy/dx)  Cos^{3}x = 0
 (d^{2}y/dx^{2}) + x(dy/dx)^{3} = 0
In the above differential equations, the degrees of the equations are three, two, and one respectively.
Related Topics
The following topics help in a better understanding of the order of the differential equations.
Example on Order and Degree of Differential Equation

Example 1: Find the order and degree of the following differential equations.
(a). 4(d^{3}y/dx^{3})  (d^{2}y/dx^{2})^{3 }+ 5(dy/dx) + 4 = 0
(b). 7(d^{4}y/dx^{4})^{2} + 5(d^{2}y/dx^{2})^{4}+ 9(dy/dx)^{8} + 11 = 0
(c). 3(d^{2}y/dx^{2}) + x(dy/dx)^{3} = 0
(d). (y''')^{2 }+ x^{2}(y')^{3}  2x + 11 = 0
(e). (dy/dx)^{2} + (dy/dx)  Cos^{3}x = 0
Solution:
(a). The differential equation is of order three, and the degree one.
(b). The differential equation is of fourthorder and second degree.
(c). This differential equation is of secondorder, and first degree.
(b). The differential equation is of the third order and second degree.
(c). The differential equation is of the first order and second degree.
Practice Questions on Order and Degree of Differential Equation
FAQs on Order and Degree of Differential Equation
How Do You Find the Order and Degree of Differential Equation?
The order of a differential equation can be found by identifying the highest derivative which can be found fin the differential equation. And the degree of the differential equation is the power of this highest order derivative in the differential equation. Let us check for the order and degree of the differential equation from the following examples of differential equations.
dy/dx = e^{x}, (d^{4}y/dx^{4}) + y = 0, (d^{3}y/dx^{3})^{2} + x^{2}(d^{2}y/dx^{2}) + xdy/dx + 3= 0
In these differential equations, the differential equations are of first order  first degree, fourth order  first degree, and third order  second degree respectively.
What Is the Order and Degree of Differential Equation?
The degree of the differential equation is the power of the highest ordered derivative present in the equation. The order of the differential equation is to be first identified, to find the degree of the differential equation. To find the degree of the differential equation, we need to have a positive integer as the index of each derivative.
How Do You Know If the Differential Equation Is A First Order or Second Order Differential Equation?
The firstorder differential equation has a degree equal to 1. All the linear equations in the form of derivatives are of the first order. It has only the first derivative dy/dx. And the equation which includes the secondorder derivative is the secondorder differential equation. It is represented as; d/dx(dy/dx) = d^{2}y/dx^{2} = f”(x) = y”
What Is the Difference Between the Order and Degree of Differential Equation?
The order of the differential equation is different from the degree of the differential equation. The order of the differential equation is the highest derivative in the differential equation and the degree of the differential equation is the power of this highest derivative in the differential equation. The order of the differential equation is to be identified first, before finding the degree of the differential equation.
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