Order and degree of a differential equation
The order is the order of the highest derivative in the equation. The degree is the highest power of that derivative, defined only for a polynomial in the derivatives.
How to find the order and degree
- 1List every derivative in the equation: y′, y″, y‴, ...
- 2The highest one gives the order. If y″ appears and y‴ does not, the order is 2.
- 3Check that the DE is a polynomial in its derivatives. If any derivative is inside sin, cos, tan, log or e^( ), the degree is not defined.
- 4Extra help (not in your textbook): if a derivative has a fractional power, raise both sides to a suitable power first. If a derivative is in a denominator, multiply through first.
- 5The degree is now the highest power of the highest order derivative.
A worked set
| Differential equation | Order | Degree | Note |
|---|---|---|---|
| dy/dx = cos x | 1 | 1 | cos x holds x, not a derivative |
| (d²y/dx²)³ + (dy/dx)⁴ + y = 0 | 2 | 3 | Use the power of y″, not of y′ |
| y‴ + 2y″ + y′ = 0 | 3 | 1 | Highest derivative y‴ has power 1 |
| d²y/dx² + 2(dy/dx) + sin y = 0 | 2 | 1 | sin y is allowed; it has no derivative inside |
| y″ + cos(y′) = 0 | 2 | Not defined | y′ is inside cos |
| [1 + (dy/dx)²]^(3/2) = d²y/dx² | 2 | 2 | Extra help (not in your textbook): square both sides to get [1 + (y′)²]³ = (y″)²; the board's answer is degree 2 |
Order vs degree
Find the order first. The degree is always read from the highest order derivative, never from a lower one.
Example 1: direct
Find the order and degree (if defined) of: (i) x(y″)³ + (y′)⁵ − y = 0, (ii) y‴ + x²y″ + e^(y′) = 0.
- (i) The highest derivative is y″, so the order is 2.
- (i) The DE is a polynomial in y′ and y″. The power of y″ is 3, so the degree is 3.
- (ii) The highest derivative is y‴, so the order is 3.
- (ii) The term e^(y′) has a derivative inside an exponential. The DE is not a polynomial in its derivatives.
(i) Order 2, degree 3. (ii) Order 3, degree not defined.
Example 2: extra help (not in your textbook), fractional powers
Find the order and degree of √(1 + (dy/dx)²) = (d²y/dx²)^(1/3).
- The highest derivative is d²y/dx², so the order is 2.
- The powers 1/2 and 1/3 are not whole numbers. Raise both sides to the power 6 (the LCM of 2 and 3).
- [1 + (dy/dx)²]³ = (d²y/dx²)²
- Now the DE is a polynomial in its derivatives. The highest power of d²y/dx² is 2.
Order = 2, degree = 2.
Traps examiners set
- The degree comes from the highest order derivative, not the biggest power anywhere. In (y′)⁴ + y″ = 0 the degree is 1.
- sin y, eʸ or log x do not spoil the degree. Only a derivative inside such a function does.
- Extra help (not in your textbook): with a derivative in a denominator, multiply through first. y′ + 1/y′ = 3 becomes (y′)² − 3y′ + 1 = 0, so the degree is 2.
- Order and degree are never zero, negative or fractions.
Engineers design the suspension of an auto-rickshaw or a scooter with second order DEs. The bounce depends on acceleration, which is a second derivative.
1 marks: “Find the sum of the order and the degree of (d²y/dx²)³ + (dy/dx)⁴ − 2y = 0.” Highest derivative d²y/dx², so order 2. Its power is 3, so degree 3. Sum = 2 + 3 = 5.
1 marks: “Find the degree of y‴ + (y″)² + sin(y′) = 0.” y′ is inside sin, so the DE is not a polynomial in its derivatives. The degree is not defined (the order is 3).
2 marks: “Find the order and degree (if defined) of (i) y(d²y/dx²) + (dy/dx)³ = x, (ii) (d³y/dx³)² = sin(d²y/dx²).” (i) Highest derivative d²y/dx², so order 2. It is a polynomial in the derivatives and d²y/dx² has power 1, so degree 1. (ii) Order 3. d²y/dx² is inside sin, so the degree is not defined.
Order: how high? Degree: how strong? Order is the highest derivative; degree is its power, and only for a polynomial.