Question

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

Short answer

The order is 1.

Step-by-step solution

Identifying Derivative Terms

Look at the given differential equation: \( y^{\prime} + y = 3y^{2} \). Identify all the derivative terms present in the equation.

Identifying the Order of the Derivative

In the equation \( y^{\prime} + y = 3y^{2} \), the only derivative term is \( y^{\prime} \). This represents the first derivative of \( y \) with respect to an independent variable.

Determining the Order of the Differential Equation

The order of a differential equation is the highest order of the derivative present in the equation. Here, the highest order derivative is \( y^{\prime} \), which is the first derivative.

Key concepts

First Order Differential Equations

A first order differential equation is a type of differential equation that involves derivatives of a function up to the first order only. In other words, when an equation includes just the first derivative but no higher derivatives, it is a first order differential equation. This means that the highest derivative present is the first derivative.

For instance, consider the equation given in the original problem: \( y^{\prime} + y = 3y^{2} \)

• The term \( y^{\prime} \) represents the first-order derivative of \( y \) with respect to the variable, usually time or space.
• Since there is no second or higher-order derivative in the equation, it confirms that it is indeed a first order differential equation.

Understanding that an equation is first order helps in employing suitable mathematical techniques for finding its solutions. There are specific methods such as separation of variables or integrating factors, which are typically used for solving such equations.

Derivatives

Derivatives are a fundamental concept in calculus that represent the rate of change or slope of a function at a particular point. When we talk about derivatives in the context of differential equations, we often refer to how one quantity changes with respect to another.

For the equation in our exercise:

• \( y^{\prime} \) is the derivative of \( y \), expressing how \( y \) changes with regard to another variable, commonly seen as time (\( t \)) or position (\( x \)).

Derivatives are essential since they help models describe physical and theoretical situations in many fields like physics, engineering, or economics.
They highlight the relationship between variables, indicating how one variable's change impacts another. Calculating and understanding derivatives allows us to predict outcomes, optimize functions, and solve equations.

Order of Differential Equation

The 'order' of a differential equation refers to the highest order of the derivatives present in the equation. In simple terms, it's the level of differentiation applied to the function in the equation.

For our given example:\( y^{\prime} + y = 3y^{2} \)

• The highest order derivative here is \( y^{\prime} \).
• This term is a first-order derivative which tells us that the overall equation is of first order.

Determining the order is crucial because it influences the approach used to find solutions. Different orders of differential equations are solved by different methods and have diverse applications.
Higher-order equations get more complex, often requiring more sophisticated mathematical techniques to solve them. Identifying the order right at the start of solving can streamline how you address the problem effectively.