Degree of a Differential Equation

Concept First: What Does "Degree" Mean for a Differential Equation?

Before we define degree, we must understand a crucial restriction. The degree of a differential equation is only defined when the equation is a polynomial equation in its derivatives. This means the derivatives (like $y'$, $y''$, $y'''$, etc.) must appear only with non-negative integer exponents, and they cannot be inside functions like sine, cosine, exponential, or logarithmic functions.

For example, consider these three equations:

1. $\left( \frac{d^3y}{dx^3} \right)^2 + \left( \frac{d^2y}{dx^2} \right)^3 - \frac{dy}{dx} + y = 0$ ... (9)
2. $\left( \frac{d^2y}{dx^2} \right)^2 + \sin\left(\frac{dy}{dx}\right) - y = 0$ ... (10)
3. $\sin\left(\frac{dy}{dx}\right) + \frac{dy}{dx} = 0$ ... (11)

Equation (9) is a polynomial in $y'''$, $y''$, and $y'$ because each derivative is raised to a positive integer power. Equation (10) is a polynomial in $y''$ (the term $\left( \frac{d^2y}{dx^2} \right)^2$ is fine), but it is not a polynomial in $y'$ because $y'$ appears inside a sine function. Equation (11) is not a polynomial in $y'$ for the same reason.

Definition of Degree

When a differential equation is a polynomial equation in its derivatives, we can define its degree.

Let's break this down:

• Highest order derivative: First, find the derivative of the highest order present in the equation (e.g., $y''$, $y'''$).
• Polynomial equation: The equation must be a polynomial in all its derivatives.
• Highest power: Look at the term containing the highest order derivative. The exponent (power) of that derivative is the degree of the differential equation.

Applying the Definition: Worked Illustrations

Let's see how this definition applies to the equations mentioned in the textbook.

Illustration 1: Equation (9)

• Order: The highest order derivative is $\frac{d^3y}{dx^3}$, so the order is 3.
• Polynomial? Yes, it is a polynomial in $y'''$, $y''$, and $y'$.
• Degree: The highest order derivative is $y'''$. Its power in the equation is 2. Therefore, the degree is 2.

Illustration 2: Equation (10)

• Order: The highest order derivative is $\frac{d^2y}{dx^2}$, so the order is 2. …