Order and degree of a differential equation
NCERT Class 12 Mathematics Chapter 9 Section 9.2
Quick answer
The order and degree of differential equations both describe the highest order derivative present: the order is the order of that derivative, 3 for y‴, and the degree, defined only when the equation is a polynomial in its derivatives, is the highest power of that derivative. So (y‴) + (y″) + y = 0 has order 3 and degree 2, while y′ + sin (y′) = 0 has order 1 and no degree.
With your textbook
NCERT Mathematics Part II (Class 12), Chapter 9 “Differential Equations”, Section 9.2 “Basic Concepts”, pages 300–304.
Open the official NCERT chapter (PDF, 3.4 MB, ncert.nic.in) (opens in a new tab)
Checked against the 2026-27 reprint on .
What is a differential equation?
Differential equation
In general, an equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation.
NCERT Mathematics Part II (Class 12), Section 9.2, page 301
- With only one independent variable it is an ordinary differential equation; with several, a partial one. NCERT studies only ordinary ones and calls them simply differential equations (page 301).
- NCERT writes dy/dx = y′, d2y/dx2 = y″, d3y/dx3 = y‴, and yn for the nth derivative (Note, page 301).
In plain words: an algebraic equation such as x + y = 7 links variables only; a differential equation also contains a derivative of the unknown function. NCERT’s equation (4) on page 301, x dy/dx + y = 0, involves x, y and dy/dx. Its solutions are functions, not numbers.
Order
Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation.
NCERT Mathematics Part II (Class 12), Section 9.2.1, page 301
NCERT’s examples (pages 301–302): dy/dx = ex has order 1; y″ + y = 0 has order 2; and y‴ + x2(y″)3 = 0 has order 3, because the highest derivative in it is the third, whatever powers the lower ones carry.
Degree
The degree is defined only when the equation is a polynomial equation in the derivatives y′, y″, y‴, … (NCERT Section 9.2.2, page 302). Then:
Degree of a differential equation
degree = the highest power of the highest order derivative
(only when the equation is a polynomial in the derivatives)
NCERT’s examples (page 302):
- y‴ + 2(y″)2 − y′ + y = 0 is a polynomial in y‴, y″ and y′: order 3, degree 1.
- (y′)2 + y′ − sin2 y = 0 is a polynomial in y′, though not in y: order 1, degree 2. A function of y itself, such as sin2 y, does not matter.
- y′ + sin (y′) = 0 is not a polynomial in y′, because of sin (y′): its degree is not defined.
NCERT’s Note (page 302): order and degree, when defined, are always positive integers.
Solved examples
Example 1: A derivative inside a sine (NCERT Exercise 9.1, Q1, page 303)
Find the order and degree of d4y/dx4 + sin (y‴) = 0.
The highest derivative is the fourth, so the order is 4. The term sin (y‴) means the equation is not a polynomial in the derivatives.
Answer: order 4; degree not defined.
Example 2: A different variable (NCERT Exercise 9.1, Q3, page 303)
Find the order and degree of (ds/dt)4 + 3s d2s/dt2 = 0.
Here s depends on t. The highest derivative is d2s/dt2, which appears to the power 1; the fourth power of the first derivative does not decide the degree.
Answer: order 2; degree 1.
Example 3: Powers of every derivative (NCERT Exercise 9.1, Q6, page 303)
Find the order and degree of (y‴)2 + (y″)3 + (y′)4 + y5 = 0.
The highest derivative is y‴, of order 3, and its highest power is 2.
Answer: order 3; degree 2.
Example 4: A sine of y, not of a derivative (NCERT Exercise 9.1, Q10, page 304)
Find the order and degree of y″ + 2y′ + sin y = 0.
sin y is a function of y, not of a derivative, so the equation is still a polynomial in y″ and y′.
Answer: order 2; degree 1.
Example 5: Choose the correct answer (NCERT Exercise 9.1, Q11, page 304)
The degree of (y″)3 + (y′)2 + sin (y′) + 1 = 0 is (A) 3; (B) 2; (C) 1; (D) not defined.
The term sin (y′) is not a polynomial in y′. The cube on y″ does not help: the degree is defined only for an equation that is a polynomial in all its derivatives.
Answer: (D).
Common mistakes to avoid
Common mistake
Taking the highest power anywhere
The degree is the power of the highest order derivative, not the highest power in the equation. In Example 2, (ds/dt)4 has power 4, but the degree is 1.
Common mistake
Reading the degree off a non-polynomial equation
If any derivative sits inside sin, cos, log or an exponential, as in y‴ + y2 + ey′ = 0 (NCERT Example 1 (iii), page 303), the degree is not defined, although the order (3) is. Beyond the NCERT text: if the equation can be rewritten with the derivative freed, read the degree from that form, as with radicals in Q7: log (y′) = x is y′ = ex, of degree 1.
Common mistake
Counting functions of y as a problem
sin y or ey do not stop the degree being defined; only functions of the derivatives do (Example 4).
Quick revision summary
- A differential equation contains a derivative of the dependent variable; NCERT treats only ordinary ones, with one independent variable.
- Order: the order of the highest derivative present.
- Degree: defined only for a polynomial equation in the derivatives; it is the highest power of the highest order derivative. Both are positive integers.
- For example, (y‴) + (y″) + y = 0 has order 3 and degree 2, while y′ + sin (y′) = 0 has order 1 and no degree.
Formula sheet
- order = order of the highest derivative
- degree = highest power of the highest order derivative
- degree not defined if a derivative stays inside sin, cos, log or exp
Questions students ask
What is the difference between the order and the degree of a differential equation?
The order of a differential equation says which derivative is the highest, while the degree is the highest power to which that derivative is raised: the order counts differentiations, the degree is an exponent. The order always exists, but the degree only when the equation is a polynomial in its derivatives (NCERT Section 9.2.2, page 302), and in NCERT’s usual regular equations the order, rather than the degree, gives the number of independent arbitrary constants the general solution has. In x(y″)4 + yy′ = 0 the highest derivative is y″, raised to the power 4, so the order is 2 and the degree 4, and a question asking for their sum or product wants 6 or 8.
Why is the degree not defined when a derivative stays inside sin, cos, log or an exponential?
The degree is not defined when a derivative stays inside sin, cos, log or an exponential because the degree is read from a polynomial in the derivatives, and a polynomial has a highest power while such a term has none. Beyond the NCERT text, the series sin u = u − u3/6 + … shows this for sin (y′): written in powers of y′ it needs y′, (y′)3, (y′)5 and so on without end, and cos, log and the exponential cannot be written as polynomials either. The order needs no such form, as it asks only which derivative is highest, so y′ + sin (y′) = 0 has order 1 and no degree (NCERT page 302).
Can the degree of a differential equation be zero, negative or a fraction?
No, the degree of a differential equation, when it is defined, is always a positive integer, and so is the order (NCERT’s Note, page 302). Degree 0 is impossible because the highest derivative present carries a power of at least 1, and order 0 would mean no derivative at all, which is not a differential equation. Beyond the NCERT text, a negative or fractional power of a derivative is cleared before the degree is read, since the equation must first be a polynomial in its derivatives: y′ + 3/y″ = 0, multiplied by y″, becomes y′y″ + 3 = 0, of order 2 and degree 1, and a root can be removed by raising both sides to a suitable power. Retain the original restrictions: the rational example requires y″ ≠ 0, while squaring a radical equation may introduce an extra sign branch. Degree is read from the prescribed polynomial form, not by arbitrarily squaring an equation to increase its degree.
Is the degree of a homogeneous differential equation zero?
No, a homogeneous differential equation dy/dx = F(x, y) has degree 1, because dy/dx appears to the first power. The zero in NCERT’s definition is the degree of the function F, which is homogeneous of degree zero when F(λx, λy) = F(x, y) for every nonzero λ (NCERT pages 312–313). For example, xy dy/dx = x2 + 2y2 is homogeneous, as F = x/y + 2y/x is unchanged when x and y are both multiplied by λ, yet it has order 1 and degree 1. All three types in NCERT Section 9.4 have the form dy/dx = F(x, y), so all have order 1 and degree 1.
What is the difference between a difference equation and a differential equation?
Beyond the NCERT text, a difference equation links the terms of a sequence, whose index n moves in whole steps, while a differential equation links a function of a continuous variable with its derivatives. For example, an+1 = 2an with a1 = 3 gives the geometric progression 3, 6, 12, …, while dy/dx = 2y has the solutions y = Ce2x. A difference equation has an order too, the number of steps it spans, so the Fibonacci recurrence an = an−1 + an−2 has order 2.
Practice questions
Try each question first, then open the answer to check your working.
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Q1. Find the order and degree of y′ + 5y = 0 and of (y″)2 + cos (y′) = 0 (NCERT Exercise 9.1, Q2 and Q4, page 303).
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The first has order 1 and degree 1. The second has order 2, and its degree is not defined, because of cos (y′).
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Q2. Find the order and degree of d2y/dx2 = cos 3x + sin 3x (NCERT Exercise 9.1, Q5, page 303).
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Order 2, degree 1: the right side is a function of x only.
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Q3. Find the order and degree of y‴ + 2y″ + y′ = 0 and of y′ + y = ex (NCERT Exercise 9.1, Q7 and Q8, pages 303–304).
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Order 3, degree 1; and order 1, degree 1.
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Q4. The order of 2x2 d2y/dx2 − 3 dy/dx + y = 0 is (A) 2; (B) 1; (C) 0; (D) not defined (NCERT Exercise 9.1, Q12, page 304).
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The highest derivative is the second: (A).