Question

Determine the order of the following differential equations.\(y^{\prime \prime \prime}+y^{\prime \prime} y^{\prime}=3 x^{2}\)

Short answer

The order of the differential equation is 3.

Step-by-step solution

Identify the Derivatives

Examine the given differential equation \(y''' + y''y' = 3x^2\). Identify all the derivatives of the dependent variable present in the equation. In this equation, we have \(y'''\), \(y''\), and \(y'\).

Determine the Highest Order Derivative

The order of a differential equation is determined by the highest order derivative present. Among \(y'''\), \(y''\), and \(y'\), the highest order derivative is \(y'''\), which is the third derivative.

State the Order of the Equation

Since the highest order derivative in the differential equation is \(y'''\), the order of the given differential equation is 3.

Key concepts

Order of Differential Equations

The order of a differential equation is fundamental to understanding its behavior and complexity. It refers to the highest derivative in the equation. Just as in algebra, where the degree of a polynomial is determined by the highest power of its variable, the order of a differential equation is identified by the highest order derivative present.
For example, in the equation given, \(y''' + y''y' = 3x^2\), we have several derivatives: \(y''', y''\), and \(y'\). The highest derivative is \(y'''\), which is the third derivative of \(y\). This makes it a third-order differential equation.
Understanding the order of a differential equation helps in classifying the equation, predicting the number of initial conditions needed for a complete solution, and determining the complexity of solving it.

Derivatives

Derivatives play a crucial role in differential equations. They represent rates of change and are foundational to calculus. A derivative describes how a function changes as its input changes, and this concept is extended to functions of one or more variables in differential equations.
Derivatives can be first order, such as \(y'\), indicating the rate at which \(y\) changes with respect to \(x\), or higher order, such as second order \(y''\) and third order \(y'''\), which describe how the rate of change itself changes. In the given equation, we encounter three different derivatives: \(y'\), \(y''\), and \(y'''\). Each derivative adds a layer of dynamic behavior to the system.
Higher order derivatives are particularly insightful in real-world applications, such as physics and engineering, where they might represent acceleration or other higher-level rates of change.

Dependent Variable

In differential equations, the dependent variable is the variable that changes in response to alterations in the independent variable. It is the main focus of the equation since derivatives act upon it.
In the equation \(y''' + y''y' = 3x^2\), \(y\) is the dependent variable because it is the function being differentiated. The independent variable here is \(x\), upon which \(y\) depends.
Understanding the dependent variable is essential because it provides the output or result that the equation models. When solving differential equations, we seek a function \(y(x)\) that satisfies the equation. This solution function reflects how the dependent variable evolves in relation to the independent variable.