Order and Degree of a Differential Equation

This page develops a precise and practical understanding of the order and degree of a differential equation: what they mean, when they are defined, and how to compute them correctly in diverse situations. All mathematics is rendered using LaTeX via MathJax, with no custom CSS or JavaScript.

Core definitions

Order

The order of a differential equation is the highest order of derivative present in the equation. For instance:

• Example: \( y'' + 3y' - 4y = 0 \) → order \(2\).
• Example: \( x\,\frac{d^3y}{dx^3} + \sin(x)\,\frac{dy}{dx} = e^x \) → order \(3\).
• For PDEs: \( u_{xx} + u_{yy} = 0 \) → order \(2\).

Degree

The degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial in derivatives. That means:

• No fractional powers of derivatives.
• No radicals involving derivatives.
• No transcendental functions of derivatives.

Examples:

• \( (y'')^2 + y' + y = 0 \) → order \(2\), degree \(2\).
• \( y'' + (y')^3 + y = 0 \) → order \(2\), degree \(1\).
• \( \sqrt{y''} + y = 0 \) → degree undefined (not polynomial in derivatives).

Step-by-step method

1. Identify the highest derivative → gives the order.
2. Check if the equation is polynomial in derivatives.
3. If radicals/fractional powers exist, try to remove them by raising to an integer power.
4. Once polynomial form is achieved, the exponent of the highest order derivative = degree.

Worked examples

1. \( y'' + 5y' + 6y = 0 \) → order \(2\), degree \(1\).
2. \( (y'')^3 - (y')^2 + y = 0 \) → order \(2\), degree \(3\).
3. \( \sqrt{y'' + y} = x \) → square → \( y'' + y = x^2 \) → order \(2\), degree \(1\).
4. \( e^{y'} + y = x \) → order \(1\), degree undefined.
5. \( (y'')^{1/3} + y = 0 \) → cube → \( y'' = -y^3 \) → order \(2\), degree \(1\).

Practice problems

1. \( y' + y = x \)
2. \( (y'')^2 + y' + y = 0 \)
3. \( \sqrt{y''+y'} = x \)
4. \( e^{y'} + x y = 0 \)
5. \( (y^{(3)})^{4/3} + y = 0 \)