Understanding the Order and Degree of Differential Equations

Differential equations (DEs) are mathematical equations that involve derivatives of one or more functions. They are fundamental in various fields such as physics, engineering, economics, and biology. To classify and solve differential equations, it is essential to understand two key concepts: the order and the degree of a differential equation.

Order of a Differential Equation

The order of a differential equation is the highest derivative (order of the derivative) that appears in the equation. It gives us an idea of how many initial conditions are needed to uniquely determine a solution to the DE.

Examples:

1. $\frac{dy}{dx} + y = e^x$ is a first-order differential equation because the highest derivative is $\frac{dy}{dx}$ (first derivative).
2. $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - y = 0$ is a second-order differential equation because the highest derivative is $\frac{d^2y}{dx^2}$ (second derivative).

Degree of a Differential Equation

The degree of a differential equation is the power of the highest order derivative in the equation, provided that the DE is a polynomial equation in its derivatives. If the differential equation is not polynomial in its derivatives, then the degree is not defined.

Examples:

1. $\left(\frac{dy}{dx}\right)^2 - y = 0$ has a degree of 2 because the highest order derivative $\frac{dy}{dx}$ is squared.
2. $\frac{d^2y}{dx^2} - (\frac{dy}{dx})^3 + y = 0$ has a degree of 3 because the first derivative $\frac{dy}{dx}$ is raised to the third power, which is the highest power of any derivative in the equation.

Table of Differences

Aspect	Order	Degree
Definition	Highest derivative in the DE.	Power of the highest order derivative, if polynomial in derivatives.
Determination	Count the highest derivative.	Look for the highest power of the highest derivative.
Example	$\frac{d^3y}{dx^3} + y = 0$ is third-order.	$\left(\frac{d^2y}{dx^2}\right)^2 + y = 0$ has a degree of 2.
Importance	Indicates the number of initial conditions needed.	Indicates the highest power of the leading derivative term.

Formulas

The general form of a differential equation can be expressed as:

$$ F\left(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \ldots, \frac{d^ny}{dx^n}\right) = 0 $$

• The order of the DE is $n$, if $\frac{d^ny}{dx^n}$ is the highest derivative present.
• The degree is the power of $\frac{d^ny}{dx^n}$, provided the equation is polynomial in $\frac{d^ny}{dx^n}$.

Examples to Explain Important Points

Example 1: Order and Degree

Consider the differential equation:

$$ \left(\frac{d^3y}{dx^3}\right)^2 - 3\left(\frac{d^2y}{dx^2}\right)^3 + \frac{dy}{dx} - y = 0 $$

• The order of this DE is 3 because the highest derivative is $\frac{d^3y}{dx^3}$.
• The degree of this DE is 2 because the highest order derivative $\frac{d^3y}{dx^3}$ is squared.

Example 2: Undefined Degree

Now, let's look at a differential equation where the degree is not defined:

$$ e^{\frac{dy}{dx}} + y = 0 $$

• The order of this DE is 1 because the highest derivative is $\frac{dy}{dx}$.
• The degree is not defined because the equation is not polynomial in $\frac{dy}{dx}$ (it involves an exponential function of the derivative).

Example 3: Implicit Differential Equation

Sometimes, the differential equation may not be explicitly solved for the highest derivative. For example:

$$ (y'')^3 - y \cdot y' = 0 $$

Here, $y''$ denotes $\frac{d^2y}{dx^2}$.

• To find the order, we identify the highest derivative, which is $y''$, so the order is 2.
• To find the degree, we look at the power of $y''$, which is 3, so the degree is 3.

Understanding the order and degree of a differential equation is crucial for selecting the appropriate method to solve it and for understanding the behavior of its solutions. It is a fundamental step in the study of differential equations.