Question

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed. $$y^{\prime}(t)=y(2-y), y(0)=1$$

Short answer

Based on the step-by-step solution described above, provide a short answer describing how to sketch the direction field and solution curve for the differential equation y'(t) = y(2-y) with the initial condition y(0) = 1. To sketch the direction field and solution curve for the given differential equation and initial condition, first set up a coordinate system with t on the x-axis and y on the y-axis within the window -2≤t≤2, -2≤y≤2, and create a grid. Next, calculate the slope of the field at each grid point using the given equation and plot small line segments with the corresponding slopes. Finally, sketch the solution curve for the given initial condition by smoothly following the direction of the line segments, starting at the point (0,1).

Step-by-step solution

Set up the coordinate system and grid

First, we need to set up a coordinate system with t on the x-axis and y on the y-axis. We will work within the window -2\(\leq\)t\(\leq\)2, -2\(\leq\)y\(\leq\)2. Draw a grid with even spacing along both axes.

Calculate the slope

Now, for each grid point, we need to calculate the slope of the field. The given equation is y'(t) = y(2-y). For example, take the grid point (0,1). At this point, the slope will be: y'(0) = 1*(2-1) = 1. Calculate the slopes at other grid points similarly.

Draw small line segments

Using the calculated slopes, we can now draw small line segments at each grid point to represent the direction of the field. The slope of these segments should match the calculated values. Make sure each segment has the same length, as their length is not significant in a direction field. For example, at the point (0,1), draw a small line segment with a slope of 1.

Sketch the solution curve for the given initial condition

Lastly, we need to sketch the solution curve for the given initial condition y(0) = 1. Start at the point (0,1) and follow the direction field along the line segments, drawing a continuous curve that represents the solution. It should pass through (0,1) and smoothly follow the direction of the segments on the grid. With these steps completed, you now have a sketch of the direction field and the solution curve for the given differential equation and initial condition.

Key concepts

Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives. In essence, it describes how a particular quantity changes over time or space. For example, the differential equation given in our exercise, \(y'(t) = y(2-y)\), artfully connects the derivative of \(y\) with respect to \(t\), denoted \(y'(t)\), to the function \(y\) itself. When embarking on the journey of solving such equations, we must recognize the importance of understanding not just the solution but the behavior and characteristics of the function as it evolves.
One valuable quality of differential equations is they enable us to model a vast range of real-world phenomena, from physics to biology, economics to engineering. Unraveling the secrets of these equations is akin to decoding the language of the universe. In a classroom or homework setting, mastering differential equations is crucial for students aiming to pursue careers in STEM fields.

Slope Fields

Imagine a field of arrows, with each pointing you in a unique direction. This is the essence of a slope field, also known as a direction field. It is a visual representation of a differential equation at a plethora of points in the given interval. By sketching short lines with slopes determined by the differential equation at each point, you build a 'slope field' that serves as a graphical guide.
In our exercise, we determine the slope of the field at each point by using the equation \(y'(t) = y(2-y)\). This reveals the incline or decline at that specific location. A slope field allows us to predict the path of a curve even before we have the exact solution to a differential equation. For students in calculus, drawing slope fields is an engaging and insightful method to visualize solutions and predict the behavior of functions graphically.

Initial Conditions

Initial conditions act as the 'starting line' for a differential equation's solution. They pinpoint the exact location where the solution curve should begin on a slope field. In the problem provided, the initial condition is \(y(0)=1\). This tells us that when \(t=0\), the value of \(y\) is exactly 1. Such stipulations are the key to solving differential equations uniquely because without them, we might end up with a multitude of potential solutions.
With the initial condition in hand, we can trace the solution curve amidst the sea of slopes provided by the field. This curve must begin at the point dictated by the initial condition and proceed along the pre-drawn slopes, ensuring accuracy in both visual representation and mathematical understanding. Recognizing the significance of initial conditions can empower students to tackle complex problems with confidence and precision.

Calculus Problem Solving

Solving a calculus problem, particularly one revolving around differential equations and slope fields, is not just about getting to the answer—it's about understanding the process and the relationships between components. By following the steps to sketch a direction field and subsequently drawing the specific solution curve that aligns with the given initial condition, students practice integrating multiple concepts—from the mechanics of derivatives to the intuition of graphing.
Ensuring that you accurately set up coordinate systems, compute slopes judiciously, draw line segments attentively, and finally, trace the solution curve with precision, can feel like directing an orchestral performance with mathematical instruments. It's this harmonious blend of theory and application that makes calculus both challenging and exhilarating. Encouraging students to embrace each step and reflect on their significance deepens their mathematical aptitude and prepares them for more advanced problem-solving scenarios.