Ordinary Differential Equations: Order and Degree

Understanding Differential Equations

A differential equation is an equation that relates a function with its derivatives. These equations are fundamental in describing how quantities change. They are used extensively in physics, engineering, biology, economics, and many other fields to model phenomena involving rates of change.

For instance, Newton's law of cooling, which describes how the temperature of an object changes over time, is a differential equation. Similarly, population growth models and the motion of objects under force are often expressed using differential equations.

Types of Differential Equations

Differential equations are broadly classified into two types:

• Ordinary Differential Equations (ODEs): These equations involve derivatives of a function with respect to only one independent variable. For example, if $y$ is a function of $x$ alone, an ODE would involve $y$, $x$, and derivatives like $\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$, etc.
• Partial Differential Equations (PDEs): These equations involve partial derivatives of a function with respect to two or more independent variables. For example, if $u$ is a function of $x$ and $t$, a PDE might involve $\frac{\partial u}{\partial t}$, $\frac{\partial u}{\partial x}$, $\frac{\partial^2 u}{\partial x^2}$, etc.

This section focuses on Ordinary Differential Equations (ODEs).

Order of an Ordinary Differential Equation

The order of an ordinary differential equation is defined as the order of the highest derivative present in the equation. To find the order, you need to identify all the derivative terms and then determine which one is of the highest order.

Let's consider some examples to clarify this:

• The equation $\frac{dy}{dx} + 2y = x$ involves the first derivative of $y$ with respect to $x$, which is $\frac{dy}{dx}$. The highest order derivative is 1. Therefore, the order of this ODE is 1.
• The equation $\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = 0$ contains derivatives $\frac{d^2y}{dx^2}$ and $\frac{dy}{dx}$. The highest order derivative is $\frac{d^2y}{dx^2}$, which is the second derivative. Thus, the order of this ODE is 2.
• Consider the equation $y''' + (y')^2 = x$. Here, $y'''$ represents the third derivative $\frac{d^3y}{dx^3}$, and $y'$ represents the first derivative $\frac{dy}{dx}$. The highest order derivative is the third derivative. Hence, the order of this ODE is 3.

It's important to ensure that the equation is expressed in a polynomial form with respect to the derivatives. Sometimes, derivatives might appear under radicals or in fractional forms, which need to be cleared first to correctly identify the order.

Degree of an Ordinary Differential Equation

The degree of an ordinary differential equation is defined as the highest power (exponent) of the highest order derivative, after the equation has been cleared of radicals and fractions involving the derivatives.

To determine the degree, follow these steps:

1. Identify the highest order derivative in the equation.
2. Ensure the equation is free from fractional powers and radicals with respect to the derivatives. This might involve cross-multiplication or raising both sides to a suitable power.
3. Once the equation is in a polynomial form with respect to the derivatives, find the power of the highest order derivative. This power is the degree of the ODE.

Let's illustrate with examples:

• For the equation $\frac{dy}{dx} + 2y = x$:
  • The highest order derivative is $\frac{dy}{dx}$ (order 1).
  • The power of $\frac{dy}{dx}$ in the equation is 1.
  • Therefore, the degree of this ODE is 1.
• Consider the equation $\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = 0$:
  • The highest order derivative is $\frac{d^2y}{dx^2}$ (order 2).
  • The power of $\frac{d^2y}{dx^2}$ in the equation is 1.
  • Therefore, the degree of this ODE is 1.
• Now, let's look at $(\frac{dy}{dx})^3 - \frac{d^2y}{dx^2} = 5$:
  • The derivatives involved are $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$.
  • The highest order derivative is $\frac{d^2y}{dx^2}$ (order 2).
  • The power of $\frac{d^2y}{dx^2}$ in the equation is 1.
  • Therefore, the degree of this ODE is 1.
• Consider the equation $(\frac{d^2y}{dx^2})^3 + (\frac{dy}{dx})^2 = \sin(x)$.
  • The highest order derivative is $\frac{d^2y}{dx^2}$ (order 2).
  • The power of $\frac{d^2y}{dx^2}$ in the equation is 3.
  • Therefore, the degree of this ODE is 3.

Handling Radicals and Fractions

Sometimes, the derivatives might appear under a square root or in a denominator. In such cases, we must first eliminate these to determine the degree accurately.

Example: Find the order and degree of $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2y}{dx^2}$.

Step 1: Identify derivatives. We have $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$. The highest order derivative is $\frac{d^2y}{dx^2}$, so the order is 2.

Step 2: Clear radicals. The equation has a square root. To remove it, we square both sides:

$(\sqrt{1 + (\frac{dy}{dx})^2})^2 = (\frac{d^2y}{dx^2})^2$

This simplifies to:

$1 + (\frac{dy}{dx})^2 = (\frac{d^2y}{dx^2})^2$

Step 3: Determine the degree. The equation is now free of radicals. The highest order derivative is $\frac{d^2y}{dx^2}$. Its power in the equation is 2.

Therefore, the degree of this ODE is 2.

Another example: Find the order and degree of $\frac{d^2y}{dx^2} + \frac{dy}{dx} = x$.

In this equation, the highest order derivative is $\frac{d^2y}{dx^2}$, so the order is 2. The power of $\frac{d^2y}{dx^2}$ is 1. There are no radicals or fractions involving derivatives. Thus, the degree is 1.

Important Considerations for Degree

The degree is only defined for ODEs that are polynomial in the derivatives. If an equation cannot be expressed in a polynomial form with respect to its derivatives, its degree is not defined.

For example, consider the equation $\frac{dy}{dx} = e^{\frac{dy}{dx}}$. This equation cannot be written as a polynomial in $\frac{dy}{dx}$. Hence, its degree is undefined.

Similarly, for an equation like $\sin(\frac{dy}{dx}) = x$, the degree is undefined because $\sin(\frac{dy}{dx})$ cannot be expressed as a polynomial in $\frac{dy}{dx}$.

Summary Table: Order vs. Degree

Practice Problems and Tips

Solved Examples for JEE Aspirants

Example 1:

Find the order and degree of the differential equation: $x (\frac{dy}{dx})^2 + y = x \frac{dy}{dx}$

• Order: The highest derivative is $\frac{dy}{dx}$, which is of the 1st order. So, the order is 1.
• Degree: The equation is already in a polynomial form with respect to $\frac{dy}{dx}$. The term with the highest order derivative is $x (\frac{dy}{dx})^2$. The power of $\frac{dy}{dx}$ here is 2. So, the degree is 2.

Example 2:

Find the order and degree of: $\frac{d^3y}{dx^3} + 3(\frac{d^2y}{dx^2})^2 + 4\frac{dy}{dx} + y = e^x$

• Order: The highest derivative is $\frac{d^3y}{dx^3}$, which is of the 3rd order. So, the order is 3.
• Degree: The equation is polynomial in its derivatives. The highest order derivative is $\frac{d^3y}{dx^3}$, and its power is 1. So, the degree is 1.

Example 3:

Find the order and degree of: $(\frac{d^2y}{dx^2})^2 = \sqrt{1 + (\frac{dy}{dx})^2}$

• Order: The highest derivative is $\frac{d^2y}{dx^2}$, so the order is 2.
• Degree: To find the degree, we must eliminate the square root. Square both sides: $(\frac{d^2y}{dx^2})^4 = 1 + (\frac{dy}{dx})^2$ Now, the equation is polynomial. The highest order derivative is $\frac{d^2y}{dx^2}$, and its power is 4. So, the degree is 4.

Example 4:

Find the order and degree of: $\frac{dy}{dx} = \sin(x)$

• Order: The highest derivative is $\frac{dy}{dx}$, order 1.
• Degree: The power of $\frac{dy}{dx}$ is 1. So, the degree is 1.

Example 5:

Find the order and degree of: $\frac{d^2y}{dx^2} + \sin(\frac{dy}{dx}) = 0$

• Order: The highest derivative is $\frac{d^2y}{dx^2}$, order 2.
• Degree: The term $\sin(\frac{dy}{dx})$ makes the equation non-polynomial in its derivatives. Therefore, the degree is undefined.