What are the order and degree, respectively, of the differential equation \({\left( {\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}} \right)^2} = {{\rm{y}}^4} + {\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^5}?\)
3, 2
Understanding Order and Degree of a Differential Equation
The question asks us to find the order and the degree of the given differential equation. To do this, we first need to understand what order and degree mean in the context of differential equations.
What is the Order of a Differential Equation?
The order of a differential equation is determined by the highest order derivative present in the equation. For example, if the highest derivative is \(\frac{{\rm{dy}}}{{\rm{dx}}}\), the order is 1. If the highest derivative is \(\frac{{{{\rm{d}}^2}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^2}}}\), the order is 2, and so on.
What is the Degree of a Differential Equation?
The degree of a differential equation is the power of the highest order derivative term after the equation has been made free from radicals and fractions as far as the derivatives are concerned. If the equation is a polynomial in derivatives, the degree is simply the highest power of the highest order derivative.
Analyzing the Given Differential Equation
The differential equation provided is:
\({\left( {\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}} \right)^2} = {{\rm{y}}^4} + {\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^5}\)
Let's identify the derivatives present in this equation:
- \(\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}\): This is a third-order derivative.
- \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\): This is a first-order derivative.
Determining the Order
The highest order derivative present in the equation is \(\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}\). Therefore, the order of the differential equation is 3.
Determining the Degree
First, we need to check if the equation is a polynomial in terms of the derivatives and if it's free of radicals or fractions involving derivatives. The given equation is already in a form where the derivatives are raised to integer powers, and there are no radicals or fractions involving them. Thus, it is a polynomial in derivatives.
The highest order derivative is \(\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}\). Its power in the equation is 2, from the term \({\left( {\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}} \right)^2}\).
The term with the highest power of the first-order derivative \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) is \({\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^5}\), which has a power of 5. However, the degree is determined by the power of the highest order derivative, not the highest power among all derivatives.
Since the highest order derivative is \(\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}\) and its power is 2, the degree of the differential equation is 2.
Conclusion
Based on our analysis:
- The order of the differential equation is 3.
- The degree of the differential equation is 2.
Therefore, the order and degree, respectively, are 3 and 2.
| Characteristic | Value | Reasoning |
|---|---|---|
| Highest Derivative Order | 3 | The highest derivative term is \(\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}\). |
| Order of Equation | 3 | Equals the highest derivative order. |
| Power of Highest Derivative | 2 | The term \(\left( \frac{{\rm{d}}^3{\rm{y}}}{{\rm{d}}{{\rm{x}}^3}} \right)^2\) shows the power of the third-order derivative. |
| Degree of Equation | 2 | Equals the power of the highest order derivative after ensuring it's a polynomial in derivatives. |
Revision Table: Order and Degree
| Concept | Definition | How to find |
|---|---|---|
| Order | The order of the highest derivative in the equation. | Identify all derivatives and find the largest integer representing the number of differentiations. |
| Degree | The power of the highest order derivative after the equation is a polynomial in derivatives. | Ensure no radicals/fractions involve derivatives. Find the highest order derivative term and note its exponent. |
Additional Information: Types of Differential Equations
Differential equations are fundamental in various fields of science and engineering. They can be classified in several ways:
- Based on Type:
- Ordinary Differential Equations (ODEs): Involve derivatives of a function with respect to a single independent variable (like \(\frac{{\rm{dy}}}{{\rm{dx}}}\)).
- Partial Differential Equations (PDEs): Involve partial derivatives of a function with respect to multiple independent variables (like \(\frac{{\partial z}}{{\partial x}}, \frac{{\partial z}}{{\partial y}}\)).
- Based on Linearity:
- Linear Differential Equations: The dependent variable and its derivatives appear only in the first power, and there are no products of the dependent variable and its derivatives. For example, \({\frac{{{\rm{d}}^2}{\rm{y}}}{{{\rm{d}}{{\rm{x}}^2}}}} + x\frac{{\rm{dy}}}{{\rm{dx}}} + y = \sin(x)\).
- Non-linear Differential Equations: Do not satisfy the conditions for a linear equation. The given equation \({\left( {\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}} \right)^2} = {{\rm{y}}^4} + {\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^5}\) is non-linear because of the terms \({\left( {\frac{{{{\rm{d}}^3}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^3}}}} \right)^2}\), \( {{\rm{y}}^4}\), and \({\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^5}\) (powers greater than 1 or products of the dependent variable).
Understanding the order and degree is often the first step in classifying and determining methods to solve a differential equation.
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