Question

Slope Field In Exercises \(67-72,\) use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition. $$ \frac{d y}{d x}=\frac{1}{2} e^{-x / 8} \sin \frac{\pi y}{4}, \quad y(0)=2 $$

Short answer

The solution involves graphing a slope field for the given differential equation, then using the given initial condition to individually graph the solution satisfying it.

Step-by-step solution

Compute the Slope Field

Use a computer algebra system to graph the slope field for the given differential equation. The slope field of a differential equation provides the slope of the solution curve at any point in the plane. For \(\frac{d y}{d x}=\frac{1}{2} e^{-x / 8} \sin \frac{\pi y}{4}\), you can plot this using a program like MATLAB or Wolfram Alpha. For every point in the plane, calculate the slope using the equation and represent it as a little line segment.

Identify the Initial Condition

The initial condition provided in the exercise is \(y(0) = 2\). This means that when \(x = 0\), \(y = 2\). This will be the starting point for graphing the solution curve.

Graph the Solution

Use the initial condition \(y(0) = 2\) to graph the solution of the differential equation. In the slope field, start at the point \(x = 0\), \(y = 2\), and follow the slope indicated by the little line segments. This will lead you to the solution curve. You can do this using a computer algebra system.

Key concepts

Differential Equation

A differential equation is a mathematical equation that relates some function with its derivatives. In this particular case, our focus is on a first-order differential equation: \( \frac{d y}{d x}=\frac{1}{2} e^{-x / 8} \sin \frac{\pi y}{4} \). This equation involves a rate of change of \( y \) with respect to \( x \).
A slope field is a visual representation of this differential equation, showing tiny line segments that indicate the slope of the solution curve at various points throughout the plane. Each of these segments gives us insight into the behavior of potential solutions, making it easier to intuitively understand how the solutions may evolve over a defined region.
Differential equations like the one given here are commonly used in fields such as physics, engineering, and economics to model complex systems that change over time.

Initial Condition

An initial condition specifies a starting point for the solution of a differential equation, providing a specific value for the function at a particular point. In this exercise, the initial condition is \( y(0) = 2 \).
This means that when \( x = 0 \), \( y \) must equal 2. Such conditions are crucial because they allow us to find a unique, specific solution to an otherwise broad and general differential equation. Without these conditions, we would have infinitely many potential solution curves fitting within the slope field.

• Initial conditions are essential in problems that model real-world situations, where starting values are known, such as at time \( t = 0 \).
• Using the initial condition: We start the graph of the solution at point \( (0, 2) \) on the slope field and follow the indicated slopes to sketch the curve.

Computer Algebra System

A computer algebra system (CAS) is a software program that allows for symbolic mathematics manipulation. These systems are incredibly helpful for solving, visualizing, and manipulating mathematical equations like differential equations.
In the context of this exercise, a CAS can graph the slope field and the solution curve for the given differential equation and initial condition.

• Programs like MATLAB or Wolfram Alpha are popular choices for performing these tasks, enabling students to visualize complex mathematical concepts easily.
• The use of a CAS aids in confirming the theoretical solutions or checks by providing accurate graphical representations.
• By entering the differential equation into a CAS, students can see how the slope field looks and how the initial condition leads to a particular solution path.

This powerful tool enhances learning and provides a deeper insight into the dynamics of differential equations, making it more accessible and interactive than manual calculations.