medium difficulty9 min

Order and Degree of a Differential Equation

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.

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

Practice This Concept

Understand it

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.

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.

Formula and rule

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.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Example: Find order and degree of √(d2y/dx2) + dy/dx = x.

1Squaring gives d2y/dx2 = (x - dy/dx)².
2Now the highest derivative is d2y/dx2 and its power is 1.
3Hence order = 2 and degree = 1, after converting to polynomial form.

Method you can reuse

1Identify all derivatives present.
2Pick the derivative with highest order.
3If degree is asked, first make the equation polynomial in derivatives if valid.
4Then read the power of the highest order derivative.
5State clearly when degree is not defined.

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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.

Common wrong answer

Writing degree = 2 for √(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.

Don't mix this up

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.

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.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

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.

1
Enough for 1 mark
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
Add for 2 marks
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.
3
Add for 3 marks
Degree needs more care: first remove radicals, negative powers, and fractional powers of derivatives wherever possible so that derivatives occur polynomially.

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