Question

(a) Determine and sketch the isoclines \(f(t, y)=c\) for \(c=-1,0\), and 1 . (b) On each of the isoclines drawn in part (a), add representative direction field filaments. $$ y^{\prime}=-y+1 $$

Short answer

Question: Determine and sketch the isoclines for the given differential equation \(y' = -y + 1\) for \(c = -1, 0, 1\), and add representative direction field filaments on each of the isoclines. Answer: The isoclines for \(c=-1, 0, 1\) are respectively \(y=2, y=1,\) and \(y=0\). Along \(y=2\), the direction field filaments have a slope of \(-1\); along \(y=1\), they have a slope of \(0\); and along \(y=0\), they have a slope of \(1\).

Step-by-step solution

Determine the isoclines for c=-1,0 and 1

To determine the isoclines for \(c=-1,0\) and \(1\), we need to solve \(f(t, y) = c\) for each value of \(c\). In our given differential equation, \(f(t, y) = y' = -y + 1\). So we need to solve the equation \(-y + 1 = c\) for each value of \(c\). For \(c = -1\): \(-y + 1 = -1 \Rightarrow y = 2\) For \(c = 0\): \(-y + 1 = 0 \Rightarrow y = 1\) For \(c = 1\): \(-y + 1 = 1 \Rightarrow y = 0\) The isoclines for \(c=-1,0,1\) are respectively \(y=2, y=1,\) and \(y=0\).

Sketch the isoclines

On the t-y plane, sketch three horizontal lines corresponding to the isoclines determined in Step 1: - \(y = 2\) for \(c = -1\) - \(y = 1\) for \(c = 0\) - \(y = 0\) for \(c = 1\) These lines divide the t-y plane into four horizontal bands.

Add representative direction field filaments

To sketch the direction field filaments, we need to compute the slope at various points along the isoclines. Since the isoclines are horizontal lines, the slope at any point will be the same for each isocline. Along each isocline, draw small line segments with the corresponding slope at different points: - On \(y = 2\) (c = -1): The slope \(y' = -y + 1 = -2 + 1 = -1\). Draw small line segments with a slope of \(-1\). - On \(y = 1\) (c = 0): The slope \(y' = -y + 1 = -1 + 1 = 0\). Draw small horizontal line segments. - On \(y = 0\) (c = 1): The slope \(y' = -y + 1 = 0 + 1 = 1\). Draw small line segments with a slope of \(1\). Once you finish drawing the direction field filaments on the isoclines, you can observe the general behavior of the solutions to the given differential equation \(y' = -y + 1\).

Key concepts

Isoclines

Isoclines are an invaluable tool for visualizing the solutions of differential equations. They are essentially curves on which the slope of the solution curve is constant. In the context of our exercise, to find the isoclines for the given differential equation \(y' = -y + 1\), we are setting the value of the derivative \(y'\) equal to some constant \(c\).

For instance, when \(c = -1\), we are looking for all points \((t, y)\) where the slope of the tangent to the solution curve is -1, which gives us the line \(y = 2\). Similarly, for \(c = 0\), we find \(y = 1\), and for \(c = 1\), we find \(y = 0\). By plotting these horizontal lines, or isoclines, we lay the foundation for our direction field and begin to see a pattern in how the solutions behave across different values of \(y\).

Differential Equations

Differential equations, like \(y' = -y + 1\) from our exercise, are equations involving derivatives which describe the rate of change of quantities. They play a central role in mathematically expressing physical, biological, and economic processes.

Understanding differential equations involves finding the function \(y(t)\) that satisfies the given equation for the derivative \(y'\). To solve the equation, we often look at specific values of \(y'\), known as isoclines, and use these to sketch the general solution without finding the explicit function. This approach is especially useful for visual learners and enhances our grasp of the equation's implications on the behavior of the system it represents. In real-life problems, such methods can be applied to model and predict behavior in a variety of dynamic systems.

Slope Fields

Slope fields, also known as direction fields, are graphical representations that show the direction of the solution curves for differential equations at a given point in the plane. Each small segment, or filament, in a slope field indicates the slope of the solution curve passing through that point.

In creating slope fields for the differential equation \(y' = -y + 1\), one must draw these segments consistently at various points, adhering to the computed slope for different \(y\) values. These filaments serve as a visual guide aiding in predicting the paths of solution curves. By inspecting the direction field, students can quickly ascertain whether the solutions are increasing or decreasing, and how they behave as time progresses, which brings us to a deeper understanding of the solution behavior of the given differential equation.

Solution Behavior

The behavior of solutions to a differential equation tells us how the dependent variable evolves over time or some other independent variable. In analyzing the solution behavior for our exercise, after plotting the direction field filaments, we note several key trends.

For example, on the isocline where \(c = -1\), the solutions tend to decrease since the slope is negative, while on the isocline where \(c = 1\), the solutions tend to increase. When \(c = 0\), the solutions are constant, indicating an equilibrium solution. By examining these trends, one gains insight into the stability of the solutions and the equilibrium points, which are pivotal concepts in understanding the dynamics of systems modeled by differential equations.