Understanding Order and Degree of Differential Equations

Differential equations are equations that involve derivatives of a function. Two fundamental characteristics of a differential equation are its order and its degree. Let's analyze the given statements about the order and degree of two different differential equations.

Analyzing Statement 1: Degree of the first differential equation

Statement 1 says: The degree of the differential equation \(\frac{\text{dy}}{\text{dx}} + \cos \left(\frac{\text{dy}}{\text{dx}}\right) = 0\) is 1.

Let's look at the differential equation:

\(\frac{\text{dy}}{\text{dx}} + \cos \left(\frac{\text{dy}}{\text{dx}}\right) = 0\)

Order: The order of a differential equation is the order of the highest derivative present in the equation. In this equation, the highest derivative is \(\frac{\text{dy}}{\text{dx}}\), which is a first-order derivative. So, the order of this differential equation is 1.

Degree: The degree of a differential equation is the highest power of the highest order derivative present in the equation, provided the equation can be expressed as a polynomial in the derivatives. The degree is only defined if the differential equation is a polynomial equation in its derivatives.

In the given equation, the term \(\cos \left(\frac{\text{dy}}{\text{dx}}\right)\) involves the derivative \(\frac{\text{dy}}{\text{dx}}\) inside a cosine function. This means the equation cannot be expressed as a polynomial in terms of the derivatives \(\frac{\text{dy}}{\text{dx}}\) or its higher powers. Therefore, the degree of this differential equation is undefined.

Statement 1 claims the degree is 1, which is incorrect because the degree is undefined.

Analyzing Statement 2: Order of the second differential equation

Statement 2 says: The order of the differential equation \(\left(\frac{\text{d}^2\text{y}}{\text{dx}^2}\right)^3 + \cos \left(\frac{\text{dy}}{\text{dx}}\right) = 0\) is 2.

Let's look at the differential equation:

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

Order: The order of a differential equation is the order of the highest derivative present. In this equation, the derivatives present are \(\frac{\text{d}^2\text{y}}{\text{dx}^2}\) and \(\frac{\text{dy}}{\text{dx}}\). The highest order derivative is \(\frac{\text{d}^2\text{y}}{\text{dx}^2}\), which is a second-order derivative. Therefore, the order of this differential equation is 2.

Degree: Although not asked in the statement, for completeness, let's consider the degree. The highest order derivative is \(\frac{\text{d}^2\text{y}}{\text{dx}^2}\), and its power is 3. The equation is a polynomial in terms of \(\frac{\text{d}^2\text{y}}{\text{dx}^2}\) and \(\frac{\text{dy}}{\text{dx}}\), even though \(\frac{\text{dy}}{\text{dx}}\) appears inside the cosine function. Since the highest order derivative \(\frac{\text{d}^2\text{y}}{\text{dx}^2}\) appears polynomially with power 3, the degree of this differential equation is 3.

Statement 2 claims the order is 2, which is correct.

Summary of Statement Analysis

• Statement 1: The degree of \(\frac{\text{dy}}{\text{dx}} + \cos \left(\frac{\text{dy}}{\text{dx}}\right) = 0\) is 1. (Incorrect - Degree is undefined)
• Statement 2: The order of \(\left(\frac{\text{d}^2\text{y}}{\text{dx}^2}\right)^3 + \cos \left(\frac{\text{dy}}{\text{dx}}\right) = 0\) is 2. (Correct)

Based on the analysis, only Statement 2 is correct.

Revision Table: Order and Degree

Additional Information: Types of Differential Equations

Understanding the type of differential equation can also be helpful. Here are a few distinctions:

• Ordinary Differential Equation (ODE): An equation involving derivatives of one or more dependent variables with respect to a single independent variable. (Both equations in the question are ODEs).
• Partial Differential Equation (PDE): An equation involving partial derivatives of a dependent variable with respect to more than one independent variable.
• Linear Differential Equation: A differential equation is linear if the dependent variable and its derivatives appear only in the first power and are not multiplied together. Also, no transcendental functions (like trigonometric, exponential, logarithmic) of the dependent variable or its derivatives are present.
  • Example of a linear ODE: \(\frac{\text{d}^2\text{y}}{\text{dx}^2} + x\frac{\text{dy}}{\text{dx}} + y = \sin(x)\)
• Non-linear Differential Equation: A differential equation that is not linear. This can be due to powers of the dependent variable or its derivatives greater than one, products of the dependent variable and its derivatives, or transcendental functions of the dependent variable or its derivatives.
  • Both equations in the question are non-linear because of the \(\cos\left(\frac{\text{dy}}{\text{dx}}\right)\) term in both, and also the power of 3 for the second derivative in the second equation.

These classifications help in determining the methods suitable for solving different types of differential equations.