Problem 1 - Calculus Early Transcendentals

Question

What is the order of \(y^{\prime \prime}(t)+9 y(t)=10 ?\)

Step-by-Step Solution

Answer

Answer: The order of the given differential equation is 2.

Step 1: Identify the given differential equation

The given differential equation is \(y^{\prime \prime}(t) + 9y(t) = 10\).

Step 2: Recognize the highest order derivative

Observe that \(y^{\prime \prime}(t)\) is the second derivative of \(y(t)\), i.e., it represents the highest order derivative present in the given equation.

Step 3: Determine the order of the differential equation

Since the highest order derivative present in the equation is the second derivative, the order of the given differential equation is 2.

Key Concepts

Second Order Differential Equation

A second order differential equation features prominently in many mathematical contexts. It involves the second derivative of a function, which gives us information about the function's curvature or acceleration. In the example, the equation is presented as \(y''(t) + 9y(t) = 10\). Here, \(y''(t)\) is the second derivative showing how the rate of change of \(y(t)\) itself changes.

The order of a differential equation is determined by the highest derivative present in the equation. In this example, the highest derivative is \(y''(t)\), marking it as a second order.

Second order differential equations are used to model a wide array of systems, such as those in physics, engineering, and even economics, due to their capacity to represent acceleration and force.

Derivatives

Derivatives are fundamental in calculus and describe how a function changes as its input changes. In a more intuitive sense, if you think about driving a car:

• The position changes with time - that's your basic function \(y(t)\).
• The speed (or first derivative \(y'(t)\)) tells you how fast you're going.
• The acceleration (or second derivative \(y''(t)\)) tells you how your speed changes over time.

In our equation, \(y''(t)\) represents this second derivative, reflecting the rate at which the speed itself changes.

Recognizing and calculating derivatives is crucial for solving differential equations, where understanding these rate of changes helps connect abstract concepts to real-world motions and processes.

Mathematics Education

Mathematics education seeks to build a solid foundation in key concepts like differential equations and derivatives. These concepts are pivotal as they often describe natural phenomena and solve real-life problems.

Engaging with second order differential equations such as \(y''(t) + 9y(t) = 10\) enhances students' understanding of dynamic systems. Learning through examples emphasizes the practical application.

• Visual aids such as graphs can help illustrate the behavior of functions and their derivatives.
• Using technology, like graphing calculators and computer software, reinforces the learning process.

Activities that allow experimenting with different equations enhance comprehension, making complex topics accessible and intuitive. Building these skills prepares learners not only for exams but for understanding the world through mathematical eyes.