1. Idea of Order and Degree
Every differential equation contains derivatives of some function. To understand its structure, two basic characteristics are examined: order and degree. These help classify the equation and indicate the nature of its solutions.
The order tells how many times the function has been differentiated at most. The degree tells how the highest-order derivative appears, but only when the equation is polynomial in all its derivatives.
1.1. Why these ideas matter
Order and degree give a quick sense of how complicated a differential equation is. Higher order generally means more arbitrary constants in the general solution. The degree, when defined, indicates whether the equation behaves linearly with respect to derivatives.
2. Definition of Order
The order of a differential equation is the highest order of derivative appearing in it.
For example, if the highest derivative in the equation is \(\frac{d^3y}{dx^3}\), then the equation is of third order.
2.1. Examples: Identifying order
- \(\frac{dy}{dx} + y = 0\) — order is 1
- \(\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 3y = 0\) — order is 2
- \(x^2 \frac{d^3y}{dx^3} + y = 4\) — order is 3
3. Definition of Degree
The degree of a differential equation is the power of the highest-order derivative after the equation has been made free from radicals and fractions involving derivatives.
This definition applies only when the equation is polynomial in all its derivatives. If it contains expressions like roots, fractional powers, or trigonometric functions of derivatives, the degree is not defined.
3.1. Steps to identify degree
- Ensure the equation is polynomial in derivatives.
- Remove radicals or fractional powers of derivatives, if possible.
- The degree is the exponent of the highest-order derivative.
3.2. Examples: Identifying degree
- \(\left(\frac{d^2y}{dx^2}\right)^3 + y = 0\) — highest derivative is squared? No, cubed, so degree is 3
- \(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 1\) — polynomial in derivatives; highest derivative is to power 1 → degree 1
- \(\sqrt{\frac{dy}{dx}} + y = 0\) — not polynomial in derivative → degree not defined
4. Conditions for Degree to Exist
Degree is defined only when:
- The differential equation is polynomial in all derivatives.
- No radicals of derivatives remain.
- No derivatives appear in denominators.
- No transcendental functions (such as \(\sin(\frac{dy}{dx})\), \(e^{\frac{dy}{dx}}\), etc.) occur.
4.1. Examples where degree does not exist
- \(e^{\frac{d^2y}{dx^2}} = x\)
- \(\sin\left(\frac{dy}{dx}\right) + y = 0\)
- \(\frac{d^2y}{dx^2} + \frac{1}{\frac{dy}{dx}} = 5\)
All these involve non-polynomial expressions of derivatives, so degree cannot be defined.
5. Worked Examples
These examples combine both order and degree to reinforce the concepts.
5.1. Example 1
Consider the equation
\(\left(\frac{d^3y}{dx^3}\right)^2 + 7\frac{dy}{dx} = x.\)
The highest derivative is \(\frac{d^3y}{dx^3}\), so order is 3. It appears squared and the equation is polynomial in derivatives, so degree is 2.
5.2. Example 2
Consider the equation
\(\sqrt{\frac{d^2y}{dx^2}} + y = 0.\)
The equation contains a radical of a derivative, so degree is not defined. The highest derivative is \(\frac{d^2y}{dx^2}\), so order is 2.
5.3. Example 3
Consider the equation
\(\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right)^4 + y = 0.\)
The equation is polynomial in derivatives. Highest derivative is \(\frac{d^2y}{dx^2}\) → order 2. The highest power of this derivative is 1 → degree 1.
6. Notes and Observations
Key points to remember:
- Order refers only to which derivative appears at the highest level.
- Degree refers to the power of that highest-order derivative, but only after the equation is cleared of radicals and fractions involving derivatives.
- Degree is undefined when the equation is non-polynomial in derivatives.
- Both concepts help classify differential equations and guide solution methods.