Question

What is the order of the following differential equation? $$ \left(x^{4}-3 x\right) y^{(5)}-\left(3 x^{2}+1\right) y^{\prime}+3 y=\sin x \cos x $$

Short answer

The order of the differential equation is 5.

Step-by-step solution

Identify the Derivatives

Examine the given differential equation \( (x^{4}-3 x) y^{(5)}-(3 x^{2}+1) y^{\prime}+3 y=\sin x \cos x \). Identify the derivatives present in the equation: \( y^{(5)} \) and \( y' \).

Determine the Order of Each Derivative

The order of a differential equation is determined by the highest order derivative present in the equation. Here, we identify two derivatives: \( y^{(5)} \), which is the fifth derivative, and \( y' \), which is the first derivative.

Identify the Highest Order Derivative

Since \( y^{(5)} \) is the highest order derivative in the equation, the order of the differential equation is the order of this derivative.

State the Order of Differential Equation

Therefore, the order of the given differential equation, based on the highest order derivative \( y^{(5)} \), is 5.

Key concepts

Understanding the Order of Differential Equations

In the realm of differential equations, the "order" of a differential equation is crucial. It refers to the highest derivative present in the equation. For example, if the equation includes second derivatives, third derivatives, all the way up to a fifth derivative, then the order is 5, which is the highest.
Understanding the order helps in categorizing the differential equation within a hierarchy and affects the methods you will use to solve it. For instance, first-order differential equations involve only the first derivative, while a fifth-order differential equation involves derivatives up to the fifth derivative.
The given exercise illustrates this concept. The highest derivative present in the equation is the fifth derivative referred to as \( y^{(5)} \). So, the order of the differential equation is determined as 5. This understanding is foundational when you advance to solving or modeling real-world situations where different orders of differential equations describe distinct dynamic systems.

Examining the Concept of Derivatives

Derivatives are the heart of calculus and differential equations. They represent the rate of change of a quantity. For example, in physics, velocity is the derivative of position with respect to time, indicating how the position changes over time.
When we talk about the order of derivatives, we refer to how many times the derivative is taken. So, the first derivative, often written as \( y' \), is the initial rate of change. The second derivative \( y'' \) represents the rate at which the first derivative itself changes, and so on.
In the given equation, two derivatives are present: \( y^{(5)} \) and \( y' \). Here, \( y' \) would typically denote a rate of change, like velocity, while \( y^{(5)} \) indicates a more complex rate of change. Each order of derivative unlocks deeper layers of change, offering insights into the dynamic behavior of the system described by the differential equation.

Approach to Solving Differential Equations

Solving differential equations typically involves finding a function or a set of functions that satisfy the equation. The process of finding these solutions depends significantly on the equation's order.
First-order differential equations can often be solved using simple integration and techniques such as separation of variables. As the order increases, techniques become more complex. For example, higher-order equations might require methods like undetermined coefficients or variation of parameters.
In some cases, specialized solutions and transformations such as the Laplace transform may be necessary. These techniques convert differential equations into algebraic forms, making them easier to solve.
Moreover, understanding initial conditions and boundary conditions is essential. These specify values at certain points, helping to determine constants in the solution that might not be immediately apparent from the equation alone. Approaching the given differential equation with these methods in mind allows you to find the most effective solution path based on the equation's order and nature.