What are the order and the degree respectively of the differential equation \(\rm x^2\left(\frac{d^3 y}{d x^3}\right)^2+\left(\frac{d y}{d x}\right)^4+\sin x=0\) ?
3, 2
Understanding Differential Equation Order and Degree
Let's break down how to find the order and the degree of a given differential equation. The equation provided is:
\(\rm x^2\left(\frac{d^3 y}{d x^3}\right)^2+\left(\frac{d y}{d x}\right)^4+\sin x=0\)
Defining Order of a Differential Equation
The order of a differential equation is determined by the highest order derivative present in the equation. To find the order, you need to look at all the derivative terms and identify the highest number of times the dependent variable (y) is differentiated with respect to the independent variable (x).
In our equation, we have two derivative terms:
- \(\frac{d^3 y}{d x^3}\): This is a third-order derivative.
- \(\frac{d y}{d x}\): This is a first-order derivative.
Comparing these, the highest order derivative is \(\frac{d^3 y}{d x^3}\), which is of order 3.
Therefore, the order of the given differential equation is 3.
Defining Degree of a Differential Equation
The degree of a differential equation is the power of the highest order derivative, provided that the differential equation can be expressed as a polynomial in the derivatives. You need to make sure the equation is free from radicals or fractions involving derivatives. Trigonometric functions, exponentials, or logarithms of derivatives can also prevent the degree from being defined in the usual way, as they are not polynomials in the derivatives.
Our given equation is:
\(\rm x^2\left(\frac{d^3 y}{d x^3}\right)^2+\left(\frac{d y}{d x}\right)^4+\sin x=0\)
Let's examine it:
- The terms involving derivatives are \(\left(\frac{d^3 y}{d x^3}\right)^2\) and \(\left(\frac{d y}{d x}\right)^4\).
- The equation is a polynomial in terms of the derivatives \(\frac{d^3 y}{d x^3}\) and \(\frac{d y}{d x}\). There are no roots, fractions, or transcendental functions (like sine, cosine, log) applied *to* the derivative terms themselves. The \(\sin x\) term does not involve a derivative, so it does not affect the degree.
- The highest order derivative is \(\frac{d^3 y}{d x^3}\).
- The power of this highest order derivative term, \(\left(\frac{d^3 y}{d x^3}\right)^2\), is 2.
Thus, the degree of the differential equation is the power of the highest order derivative, which is 2.
Therefore, the degree of the given differential equation is 2.
Summary of Order and Degree
Based on our analysis:
- Order: 3
- Degree: 2
The order and degree of the differential equation are 3 and 2, respectively.
| Feature | Definition | Applied to \(\rm x^2\left(\frac{d^3 y}{d x^3}\right)^2+\left(\frac{d y}{d x}\right)^4+\sin x=0\) | Result |
|---|---|---|---|
| Order | Highest order of derivative present | Highest derivative is \(\frac{d^3 y}{d x^3}\) (order 3) | 3 |
| Degree | Power of the highest order derivative (if equation is a polynomial in derivatives) | Highest order derivative is \(\frac{d^3 y}{d x^3}\). Its power is 2. The equation is a polynomial in derivatives. | 2 |
So, the order is 3 and the degree is 2.
Revision Table: Differential Equations
| Term | Meaning | Example |
|---|---|---|
| Differential Equation | An equation involving an independent variable, a dependent variable, and derivatives of the dependent variable with respect to the independent variable. | \(\frac{dy}{dx} = y + x\) |
| Order of DE | The order of the highest derivative in the equation. | In \(\frac{d^2y}{dx^2} + (\frac{dy}{dx})^3 = 0\), the highest order is 2. |
| Degree of DE | The power of the highest order derivative, provided the equation is a polynomial in derivatives. | In \(\frac{d^2y}{dx^2} + (\frac{dy}{dx})^3 = 0\), the highest order derivative is \(\frac{d^2y}{dx^2}\) with power 1. Degree is 1. In \((\frac{d^3y}{dx^3})^2 + y = 0\), highest order is 3, its power is 2. Degree is 2. |
Additional Information: Types of Differential Equations
Differential equations can be classified in several ways:
- Based on Type:
- Ordinary Differential Equation (ODE): Involves derivatives of a dependent variable with respect to a single independent variable (like our example).
- Partial Differential Equation (PDE): Involves partial derivatives of a dependent variable with respect to two or more independent variables.
- Based on Order: Classified by the highest order derivative (e.g., first-order DE, second-order DE, etc.).
- Based on Linearity:
- Linear Differential Equation: An equation is linear if the dependent variable and its derivatives appear only in the first power, and there are no products of the dependent variable and its derivatives. Also, coefficients depend only on the independent variable. Our example equation \(\rm x^2\left(\frac{d^3 y}{d x^3}\right)^2+\left(\frac{d y}{d x}\right)^4+\sin x=0\) is NOT linear because of the terms \(\left(\frac{d^3 y}{d x^3}\right)^2\) and \(\left(\frac{d y}{d x}\right)^4\).
- Non-linear Differential Equation: Any equation that is not linear.
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