The order and degree of a differential equation are distinct properties that describe its complexity. The order relates to the highest derivative present, while the degree relates to the power of the highest derivative, after the equation has been rationalized and cleared of fractions as far as derivatives are concerned.
Order of a Differential Equation
The order of a differential equation is defined as the order of the highest derivative appearing in the equation. For example, if the highest derivative is d²y/dx², the order is 2.
- First Order: Contains only the first derivative (e.g., dy/dx).
- Second Order: Contains the second derivative (e.g., d²y/dx²), and may or may not contain the first derivative.
- Third Order: Contains the third derivative (e.g., d³y/dx³), and may or may not contain lower-order derivatives.
- And so on...
Degree of a Differential Equation
The degree of a differential equation is the power to which the highest-order derivative is raised, after the equation has been made free from radicals and fractions in its derivatives. This means you must first eliminate any square roots, cube roots, or fractional exponents affecting the derivatives.
- To find the degree, the equation must be a polynomial equation in derivatives. This means that you should be able to express the equation in a form where the derivatives are raised to positive integer powers.
- If the equation cannot be expressed as a polynomial in derivatives, the degree is not defined.
Illustrative Examples
To clarify the distinction, consider these examples:
| Differential Equation | Order | Degree | Explanation |
|---|---|---|---|
| dy/dx + y = 0 | 1 | 1 | The highest derivative is dy/dx (first order), and its power is 1. |
| d²y/dx² + (dy/dx)² + y = x | 2 | 1 | The highest derivative is d²y/dx² (second order), and its power is 1. |
| (d²y/dx²)² + (dy/dx)³ + y = sin(x) | 2 | 2 | The highest derivative is d²y/dx² (second order), and its power is 2. |
| √(dy/dx) + y = x | 1 | Not Defined | The equation is not a polynomial equation in the derivative dy/dx because it contains a square root. Therefore, the degree is not defined. |
| (d³y/dx³)² + (d²y/dx²)^4 + (dy/dx)^6 + y = e^x | 3 | 2 | The highest derivative is d³y/dx³ (third order), and its power is 2. |
| d²y/dx² + sin(dy/dx) = 0 | 2 | Not Defined | The degree is not defined because the term sin(dy/dx) is not a polynomial function of the first derivative. |
Summary Table
| Feature | Order | Degree |
|---|---|---|
| Definition | Highest derivative in the equation. | Power of the highest derivative after the equation is expressed as a polynomial in derivatives (free from radicals and fractions). |
| Determining | Identify the highest-order derivative. | Identify the exponent of the highest-order derivative after the equation has been properly prepared. |
| Restrictions | None | The equation must be a polynomial equation in derivatives for the degree to be defined. |
In conclusion, while both order and degree help to classify differential equations, they represent different aspects of the equation's structure. Order focuses on the type of derivative with the maximum rate of change, while degree focuses on the power of that derivative, provided the equation meets certain criteria.
