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Maths > Differential Equations > Order and Degree of a Differential Equation
Differential Equations

Order and Degree of a Differential Equation

A differential equation is a mathematical equation that relates some function with its derivatives. In real-life applications, the functions represent some physical quantities while its derivatives represent the rate of change of the function with respect to its independent variables. Let’s study about the order and degree of differential equation.

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Owing to the variety of life around us, innumerable types of functions and hence, differential functions, to model the innumerable physical processes are possible. We can classify the differential equations in various ways, the simplest of them being on the basis of the order and degree of differential equation.

This is an important categorization because once grouped under this category, it is straightforward to find the general solutions of the differential equations. For example, The method of finding the general solution of a differential equation of the second order can be extended to find the general solution of a differential equation of the nth order.

Browse more Topics under Differential Equations

You’ll encounter this if you go for higher studies in mathematics. So let’s start with the basics of order and degree of differential equation for now!

Order and Degree of Differential Equation

General Differential Equation

The most general differential equation in two variables is –

f(x, y, y’, y”……) = c

where –

  • f(x, y, y’, y”…) is a function of x, y, y’, y”… and so on.
  • x is the independent variable.
  • y is the dependent variable.
  • y’, y”…. and so on, is the first order derivative of y, second order derivative of y, and so on.
  • c is some constant.

Order and Degree of Differential Equation

The order of a differential equation is the order of the highest order derivative involved in the differential equation. The degree of a differential equation is the exponent of the highest order derivative involved in the differential equation when the differential equation satisfies the following conditions –

  • All of the derivatives in the equation are free from fractional powers, positive as well as negative if any.
  • There is no involvement of the derivatives in any fraction.
  • There shouldn’t be involvement of highest order derivative as a transcendental function, trigonometric or exponential, etc. The coefficient of any term containing the highest order derivative should just be a function of x, y, or some lower order derivative.

If one or more of the aforementioned conditions are not satisfied by the differential equation, it should be first reduced to the form in which it satisfies all of the above conditions.  An equation has no degree or undefined degree if it is not reducible.

The determination of the degree of a given differential equation can be very tricky if you are not well versed with the conditions under which the degree of the differential equation is defined. So go through the given solved examples under the ‘Degree’ topic carefully and master the technique of calculating the degree of the given differential equation just by sheer inspection!

Know the Formation of Differential Equation whose General Solution is Given

Solved Examples for You

Find the Orders.

For a differential equation represented by a function f(x, y, y’) = 0; the first order derivative is the highest order derivative that has involvement in the equation. Thus, the Order of such a Differential Equation = 1. In a similar way, work out the examples below to understand the concept better –

  • \({ x\frac{d^2y}{dx^2} + y\frac{dy}{dx} + 4y^2 = 1 }\): Order = 2
  • \({ sin(\frac{d^3y}{dx^3}) = \frac{dy}{dx} + x }\): Order = 3
  •  For \({ \sqrt{(\frac{dy}{dx})^2 + 3y} = \frac{d^2y}{dx^2}}\): Order = 2

Find the Degrees.

  • \({ 3y^2(\frac{dy}{dx})^3 – \frac{d^2y}{dx^2} = sin(x^2)}\)

The highest order derivative involved here is of order 2, and its power = 1 in the equation. Thus, the order of the differential equation = 2, degree = 1.

  • \({ \sqrt{1 + (\frac{dy}{dx})^2} = y\frac{d^3y}{dx^3}}\)

Since this equation involves fractional powers, we must first get rid of them. On squaring the equation, we get – \({ 1 + (\frac{dy}{dx})^2 = y^2(\frac{d^3y}{dx^3})^2 }\). Now, we can clearly make out that the highest order derivative is of order 3 here i.e. order of the differential equation = 3 and since its power is 2 in the equation – the degree of the differential equation = 2.

  • \({ sin(\frac{dy}{dx}) + \frac{d^2y}{dx^2} +3x = 0}\)

Here, the highest order derivative is of order 2, and it has no involvement in any function. So, the order of the differential equation = 2, and degree = 1.

More Examples

  • \({e^{\frac{d^2y}{dx^2}} + sin(x)\frac{dy}{dx} = 1}\)

Here, the highest order derivative (order = 2) has involvement in an exponential function. Note that the exponential function can be expanded as a series to bring it to a polynomial form i.e. \({e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} ….}\).

Thus, the powers of the 2nd order derivative in the equation above will keep on varying as we incorporate more and more terms in the series expansion of the exponential function. Thus, the degree of the equation = Not Defined. The order of the equation = 2.

  • \({sin(\frac{dy}{dx})\frac{d^3y}{dx^3} – log(y) = x^2}\)

Here, the coefficient of the highest order derivative is a function only of \(\frac{dy}{dx}\), which is a lower order derivative. It thus defines the degree of the equation. Even if you expand the trigonometric sine function, you’ll get something like \({sinx = x – \frac{x^3}{3!} + \frac{x^5}{5!} – …..}\), which is a polynomial function with an infinite number of terms.

Since it is multiplied with  \({\frac{d^3y}{dx^3}}\), every term in the expansion would contain the term \({\frac{d^3y}{dx^3}}\), thus maintaining the degree of the highest order derivative constant, unlike our last example. Thus, the order of the differential equation = 3 and degree = 1.

This concludes our discussion on this topic of order and degree of differential equation.

Question 3: What is meant by order of differential equation?

Answer: Order of a differential equation refers to the order of the highest derivative that exists in the equation. Experts also call it as the differential coefficient that is present in the equation.

Question 4: What is meant by first order equation?

Answer: A first-order differential equation refers to an equation in which ƒ(x, y) happens to be a function of two variables and it can be defined on a region existing in the xy-plane. The equation is of first order due to the fact that it involves only the first derivative dy dx.

Question 5: What is meant by second order differential equation?

Answer: A second order differential equation refers to an equation that consists of an unknown function y along with its derivatives y’ and y”, and also the variable x.

Question 6: When can we say that a differential equation is linear?

Answer: Linear means that the appearance of the variable in an equation takes place only with a power of one. In case of a differential equation, it will be linear when the multiplication of the variables and their derivatives takes place by the constants.

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