Question

Determine the order of the following differential equations.\(\left(\frac{d y}{d t}\right)^{2}+8 \frac{d y}{d t}+3 y=4 t\)

Short answer

The order of the differential equation is 1.

Step-by-step solution

Identifying the Derivatives

First, examine the given differential equation: \( \left(\frac{d y}{d t}\right)^{2} + 8 \frac{d y}{d t} + 3y = 4t \). Determine what derivatives are involved in this differential equation. Here, we see \( \frac{d y}{d t} \), the first derivative of \( y \) with respect to \( t \).

Determining the Highest Order Derivative

After listing all derivatives, identify the one with the highest order. In the equation, only the first order derivative \( \frac{d y}{d t} \) appears, and is squared, which does not affect the order. Hence, the highest order derivative is still \( \frac{d y}{d t} \), which is the first order derivative.

Conclusion: Finding the Order of the Differential Equation

The order of a differential equation is determined by the highest derivative present. Since the highest order derivative present in the differential equation is \( \frac{d y}{d t} \) (which is the first derivative), the order of this differential equation is 1.

Key concepts

Order of Differential Equations

In calculus, the order of a differential equation is an important concept that helps us understand the complexity of the equation. The order is defined as the highest derivative present in the equation. For example, in the differential equation \( \left(\frac{d y}{d t}\right)^{2} + 8 \frac{d y}{d t} + 3y = 4t \), we identify all the derivatives by looking for the terms where derivatives appear.

• The term \( \frac{d y}{d t} \) is the first derivative of \( y \) with respect to \( t \).
• Even though this term is squared, it does not change the order of the differential equation.

Thus, the highest derivative is the first one (\( \frac{d y}{d t} \)), making this a first-order differential equation. Understanding the order is key because it affects both the method of solving the equation and the nature of its solutions.

Derivatives

Derivatives are a fundamental concept in calculus, representing the rate at which a function is changing at any given point. They are the building blocks of differential equations. In the context of a differential equation, the derivatives dictate how much one variable changes in response to another and are represented in terms such as \( \frac{d y}{d t} \), which denotes the rate of change of \( y \) with respect to \( t \).

• The notation \( \frac{d y}{d t} \) is commonly used in physics and engineering to describe things like velocity, where \( y \) could represent position and \( t \) time.
• First derivatives tell us about the slope or the incline of a function at any specific point, which often relates to velocity or speed.

Learning about derivatives allows you to understand not just how fast something changes, but also the direction and possible future changes of the system described by the function.

First Order Differential Equation

A first-order differential equation is one in which the highest derivative is the first derivative. It typically involves both the function itself and its first derivative. This type of equation is often manageable due to its relatively simple form and is common in modeling processes where change is dependent solely on the current state.

• First-order differential equations can describe a wide variety of phenomena such as exponential growth, decay, heat transfer, and fluid flow.
• They often appear in separable form, making them easier to solve by isolating each variable on different sides of the equation.
• In our original example, the first derivative \( \frac{d y}{d t} \) appears in the equation without involving any higher derivatives, confirming it is first-order.

Grasping how to identify and work with first-order differential equations lays the groundwork for understanding higher-order equations, which can model more complex systems and interactions.