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Order and Degree of a Differential Equation

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Student-friendly explanation

How to write this in exams

1
Start with the exact idea
The order of a differential equation is the order of the highest derivative present. The degree is the power of the highest order derivative after the equation is made polynomial in derivatives, if such a polynomial form is possible.
2
Then show how to use it
Identify all derivatives present. Pick the derivative with highest order. If degree is asked, first make the equation polynomial in derivatives if valid. Then read the power of the highest order derivative. State clearly when degree is not defined.
3
Add one concrete example
For (d2y/dx2)^3 + (dy/dx)^2 = x, the highest derivative is d2y/dx2, so order = 2. Its power is 3, so degree = 3.
4
Avoid this incomplete answer
Writing degree = 2 for sqrt(d2y/dx2) + dy/dx = x without first converting the equation, or taking degree from (dy/dx)^3 when d2y/dx2 is also present with power 1.

Definition

Example

For (d2y/dx2)^3 + (dy/dx)^2 = x, the highest derivative is d2y/dx2, so order = 2. Its power is 3, so degree = 3.

Rule to remember

Rule: order = highest order derivative present. Degree = exponent of the highest order derivative only after the equation is expressible as a polynomial in derivatives. If derivatives occur as sin(dy/dx), e^(dy/dx), log(dy/dx), or under an unavoidable radical, degree is not defined.

Memory hook

Order asks: how high is the derivative? Degree asks: what power has that highest derivative after cleanup?

Examples and method

Worked example

Example: Find order and degree of sqrt(d2y/dx2) + dy/dx = x. Squaring gives d2y/dx2 = (x - dy/dx)^2. Now the highest derivative is d2y/dx2 and its power is 1. Hence order = 2 and degree = 1, after converting to polynomial form.

Method to apply

Identify all derivatives present. Pick the derivative with highest order. If degree is asked, first make the equation polynomial in derivatives if valid. Then read the power of the highest order derivative. State clearly when degree is not defined.

Diagram support

A diagram is not required because this concept depends on symbolic classification of derivatives, not on a graph or geometric figure.

How CBSE asks it

Usually asked as a one-mark or two-mark classification question, sometimes with equations containing square roots or powers to test whether the student checks polynomial form before deciding degree.

Avoid common mistakes

Common confusion

Students often call the highest power of any derivative the degree. The degree must be the power of the highest order derivative, not necessarily the largest exponent appearing anywhere.

Common wrong answer

Writing degree = 2 for sqrt(d2y/dx2) + dy/dx = x without first converting the equation, or taking degree from (dy/dx)^3 when d2y/dx2 is also present with power 1.

Exam tip

Before writing degree, check whether the equation is polynomial in derivatives. If a derivative is inside a square root, denominator, or trigonometric function and cannot be converted to polynomial form, degree is not defined.

Quick check

Find the order and degree of (d2y/dx2)^2 + dy/dx = sin x.

Order = 2 because the highest derivative is d2y/dx2. Degree = 2 because the equation is polynomial in derivatives and the highest order derivative has power 2.

Answer writing and exam use

1-mark answer

2-mark answer

The order of a differential equation is the order of the highest derivative present. The degree is the power of the highest order derivative after the equation is made polynomial in derivatives, if such a polynomial form is possible. Rule: order = highest order derivative present. Degree = exponent of the highest order derivative only after the equation is expressible as a polynomial in derivatives. If derivatives occur as sin(dy/dx), e^(dy/dx), log(dy/dx), or under an unavoidable radical, degree is not defined. For (d2y/dx2)^3 + (dy/dx)^2 = x, the highest derivative is d2y/dx2, so order = 2. Its power is 3, so degree = 3.

3-mark answer

Order is read directly from the highest derivative such as dy/dx, d2y/dx2, or d3y/dx3. Degree needs more care: first remove radicals, negative powers, and fractional powers of derivatives wherever possible so that derivatives occur polynomially. If the equation cannot be expressed as a polynomial in derivatives, its degree is not defined. Rule: order = highest order derivative present. Degree = exponent of the highest order derivative only after the equation is expressible as a polynomial in derivatives. If derivatives occur as sin(dy/dx), e^(dy/dx), log(dy/dx), or under an unavoidable radical, degree is not defined. Example: Find order and degree of sqrt(d2y/dx2) + dy/dx = x. Squaring gives d2y/dx2 = (x - dy/dx)^2. Now the highest derivative is d2y/dx2 and its power is 1. Hence order = 2 and degree = 1, after converting to polynomial form. Usually asked as a one-mark or two-mark classification question, sometimes with equations containing square roots or powers to test whether the student checks polynomial form before deciding degree. Writing degree = 2 for sqrt(d2y/dx2) + dy/dx = x without first converting the equation, or taking degree from (dy/dx)^3 when d2y/dx2 is also present with power 1.

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