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Order and Degree of Differential Equations

Last Updated : 04 Sep, 2023
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Order and Degree of differential equations indicate the degree of complexity and the number of independent variables in the differential equations. The highest derivative sets the order of the equation and offers important information about the function’s behaviour and evolution. It is an important tool for dealing with scientific and engineering problems, with applications in physics, engineering, biology, and economics.

Understanding the order and degree of differential equations allows us to foresee how the function will react to changes in independent variables, allowing us to better comprehend complex systems and real-world occurrences. This inquiry delves into the significance and applications of the “Order and Degree of Differential Equations,” helping us to better comprehend the intricacies of our surroundings.

What are Differential Equations?

A differential equation is a mathematical formula that combines a function and its derivatives. The functions in real-world applications indicate physical quantities, and their derivatives show the rate at which those physical values change in relation to their independent variables. The general form of a differential equation is:

F(x, y, y’, y”, …, yn‘ ) = 0

Where,

  • x is the dependent variable,
  • y is the independent variable,
  • y’ is the first-order derivative of the function y = f(x),
  • y” is the second order derivative of the function y = f(x), and
  • . . .
  • ynis the nth order derivative of the function y = f(x).

Learn more about Differential Equation

Order of Differential Equation

The highest order of the derivative of the unknown function ‘y’ in a differential equation is referred to as the order of the equation. In other words, it is the power to which the equation’s highest derivative is raised in any given differential equation.

Example: Determine the order of Differential Equation – d3y/dx3 + 3x(dy/dx) ​= ey .

Answer:

The order of this differential equation is 3 as the highest order derivative is d3y/dx3.

First Order Differential Equation

An equation involving the first derivative of an indeterminate function is known as a first-order differential equation. It can be expressed in the following general form:

f(x, y) = dy/dx

Where,

  • f(x, y) is a known function of both x and y, and
  • dy/dx is the first derivative of the function y with respect to x.

Types of First-Order Differential Equations

There are various types of first-order differential equations, some of these are:

  • Separable Differential Equations: In separable differential equations, x and y may be separated on either side of the equation, enabling us to individually integrate either side to obtain the solution.
  • Homogeneous Differential Equations: Differential equations that have a right-hand side that can be expressed as a homogeneous function of x and y are called homogeneous differential equations. Substitution methods can be used to solve these problems.
  • Linear Differential Equations: Equations of the form dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x, are known as linear differential equations.

Second Order of Differential Equation

A second-order differential equation involves the second derivative of a function that is undetermined. It has the following representation:

f(x, y, dy/dx) = d2y/dx2

Where,

  • f(x, y, dy/dx) is a given function of x, y, and
  • dy/dx and d2y/dx2 is the second derivative of the function y with respect to x.

Types of Second-Order Differential Equations

There are various types of second-order differential equations, some of these types are:

  • Homogeneous Second-Order Differential Equations: A homogeneous function of x, y, and dy/dx can be used to describe the right-hand side of a homogeneous second-order differential equation. Substitution methods can be used to solve these problems.
  • Non-Homogeneous Second-Order Differential Equations: Second-order differential equations with a non-homogeneous term are known as non-homogeneous second-order differential equations. Several methods, such as the method of indeterminate coefficients or the method of parameter variation, can be used to solve these equations.
  • Linear Second-Order Differential Equations: Equations of the form a (d2y/dx2) + b(dy/dx) + cy = f(x), where a, b, and c are constants and f(x) is a predetermined function, are used to define linear second-order differential equations. Significant physics and engineering applications may be made of these equations.

Degree of Differential Equation

The highest derivative of an equation, taking consideration of any coefficients or constants as well, is defined as the degree of the differential equation. It is the derivative in the equation with the highest power, in other words. When working with differential equations of the polynomial kind, determining a differential equation’s degree is important.

  • For example, the equation (d2y/dx2)4+dy/dx​=3

is a second-order differential equation with degree four because the highest order derivative is d2y/dx2 and its power is 4.

Speacial Case: Degree not Defined

Most of the time we can arrange the given differential equation in the form of polynomial where exponent is the order of differentiation, but for some differential equation this can be done. Those differential equations degree is not defined.

In other words, degree of a differential equation can only be defined if and only if the polynomial function of derivatives can be generated from the differential equation.

For example, sin(d2y/dx2) + dy/dx​=0

is not a polynomial equation in derivatives because of the sine function. Therefore, its degree is not defined.

Some other examples of differential equation, where degree is not defined are:

  • (d2y/dx2)2 3(dy/dx)3 = x log(d2y/dx2)
  • x = sin(dy/dx – y cos x)
  • x – ysin x = log(dy/dx)

How To Find Order and Degree Of Differential Equation?

To find the order and degree of Differential Equation, we can use the following steps:

Step 1: Examine the Differential Equation.

Step 2: Write the Differential Equation in Standard Form.

Step 3: Determine the Order.

Step 4: Determine the Degree.

Let’s consider an example for better understanding.

Example: Find the order and degree of the following differential equation.

3(d2y/dx2)2​ – 3 = 4xdy/dx

Answer:

Step 1: Examine the Differential Equation.

After examination of the given Differential Equation, we get to know there are two derivative terms in the equation i.e., d2y/dx2, and dy/dx.

Step 2: Write the Differential Equation in Standard Form.

To write the Differential Equation in Standard Form i.e., in the decreasing order of differentiation just like polynomial.

Thus, the given equation becomes,

3(d2y/dx2)2​ – 4xdy/dx – 3 = 0

Step 3: Determine the Order.

The order of the differential equation is the highest order of derivative present. In this case, the highest order of derivative is the second derivative, so the order is 2.

Step 4: Determine the Degree.

The degree of the differential equation is determined by the highest power of the highest order derivative present. In this case, the second derivative i.e., d2y/dx2 is raised to the power of 2, so the degree is also 2.

Thus, for equation 3(d2y/dx2)2​ – 3 = 4xdy/dx, order and degree both are 2.

Examples of Order and Degree of Differential Equation

Let’s look at a few examples to better understand the order and degree of differential equations:

Example 1: dy/dx + 2x = 0

Answer:

Order: 1 (the highest derivative is the first derivative dy/dx)

Degree: 1 (the highest power of the derivative is 1)

Example 2: d2y/dx2 – 3(dy/dx)2+ 2x = 0

Answer:

Order: 2 (the highest derivative is the second derivative d2y/dx2)

Degree: 2 (the highest power of the derivative is 2)

Example 3: x2(d3y/dx3) + y3(dy/dx) = 0

Answer:

Order: 2 (the highest derivative is the second derivative d2y/dx2)

Degree: 2 (the highest power of the derivative is 2)

Solved Problems on Order and Degree of Differential Equation

Problem 1: What is the order (if defined) and degree of the differential equation: d4y/dx4 + (d2y/dx2)2 – 3(dy/dx) + y = 9?

Solution:

Since d4y/dx4 is the highest order derivative, the order of this equation is 4, and degree is 1 as the highest order derivative has exponant 1.

Problem 2: Determine the order (if defined) and degree of the differential equation: [d2y/dx2 + (dy/dx)2]4 = k2(d3y/dx3)2.

Solution:

Since d3y/dx3 is the highest order derivative, the order of this equation is 3, and degree is 2 as the highest order derivative has 2 as it’s exponant.

Problem 3: Find the order (if defined) of the differential equation: √(d2y/dx2) + dy/dx = x.

Solution:

√(d2y/dx2) + dy/dx = x

⇒ √(d2y/dx2) = -dy/dx + x

⇒ d2y/dx2 = (-dy/dx + x)2

Since d2y/dx2 is the highest order derivative and it’s exponent is 1.

Thus, the order and degree of this equation is 2 and 1 respectively.

Problem 4: What is the order (if defined) and degree of the differential equation: dy/dx + (x2 + 5)y = x5?

Solution:

Because dy/dx is the highest order derivative and it’s exponent is 1, thus this equation has an order of 1 and degree 1 as well.

Practice Problems on Order and Degree of Differential Equation

Problem: Find Order and Degree of following differential equations:

  • d3y/dx3 + 2d2y/dx2 – 5dy/dx + 6y = 0
  • 2(d2y/dx2)2 + (d4y/dx4)3 = 3dy/dx
  • d2​/dx2(ex​d3​y/dx3​) + sin(x)dy/dx ​= 0
  • (d2y/dx2)3 + d3y/dx3 = sin x

FAQs on Order and Degree of Differential Equation

1. Define the Order of a Differential Equation.

The order of a differential equation refers to the highest order of derivatives present in the equation.

2. What does a Differential Equation’s Order Represent?

Knowing the order makes it easier to ascertain how many boundary or beginning conditions are required to produce a certain solution to the problem.

3. How is the Order of a Differential Equation Determined?

Count the highest derivative that appears in the equation. For example, if the equation involves only the first derivative, it’s a first-order differential equation. If it involves the second derivative but not higher, it’s a second-order differential equation, and so on.

4. What is the Difference between a First-Order and a Second-Order Differential Equation?

A first-order differential equation involves only the first derivative of the unknown function, while a second-order differential equation involves the second derivative. Higher-order equations involve even higher derivatives.

5. Can a Differential Equation change its Order?

No, the order of a differential equation remains constant.

6. What are Some Examples of First-Order Differential Equations?

Some of the common examples of first-order differential equations are:

  • Linear first-order: dy/dx + P(x)y = Q(x)
  • Separable: dy/dx = g(x)h(y)
  • Exact: M(x, y)dx + N(x, y)dy = 0
  • Homogeneous: dy/dx = f(y/x)

7. What is the Difference between Degree and Order of a Differential Equation?

Degree refers to the highest power of derivative however order is the highest derivative’s order in a differential equation.



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