Question

Direction fields with technology Plot a direction field for the following differential equation with a graphing utility. Then find the solutions that are constant and determine which initial conditions \(y(0)=A\) lead to solutions that are increasing in time. $$y^{\prime}(t)=(y-1) \sin \pi t, 0 \leq t \leq 2,0 \leq y \leq 2$$

Short answer

Question: Determine if there are any initial conditions, y(0) = A, that lead to solutions that are increasing in time for the differential equation y'(t) = (y - 1) sin πt. Answer: No initial conditions y(0) = A lead to solutions that are increasing in time for the given differential equation.

Step-by-step solution

Plot the direction field

Use a graphing utility or online tool to plot the direction field of the given differential equation: $$y^{\prime}(t)=(y - 1) \sin \pi t$$ for \(0 \leq t \leq 2\) and \(0 \leq y \leq 2\). By observing the graph, one can get an idea of how the solutions will behave for different initial conditions.

Find the constant solutions of the differential equation

For a solution to be constant, its derivative must be equal to 0. Therefore, set the right-hand side of the differential equation equal to 0. $$(y-1) \sin \pi t = 0$$ Now, either \((y-1)\) or \(\sin \pi t\) must be equal to 0 for the equation to hold. If \(\sin \pi t = 0\), then \(t\) is an integer. However, we are looking for constant solutions, i.e., solutions that don't depend on \(t\). So, we need to consider the case where \((y-1) = 0\). Thus, \(y=1\) is a constant solution, which means that there exists a horizontal line at \(y=1\) on the direction field plot.

Determine the initial conditions that lead to increasing solutions

To determine which initial conditions \(y(0)=A\) lead to increasing solutions, we need to analyze the sign of the right-hand side of the differential equation. If the right-hand side of the equation is positive, then \(y^{\prime}(t) > 0\) and the solution is increasing at that point. Conversely, if the right-hand side is negative, the solution is decreasing. Consider the interval \(0 \leq t \leq 2\). In this interval, we have a complete period of the sine function, so \(\sin \pi t\) changes its sign from positive to negative and vice versa. Now, let's look at the other term: \((y-1)\). For \(0 \leq y < 1\), \((y-1)\) is negative. For \(y = 1\), \((y-1)\) is equal to 0. Finally, for \(1 < y \leq 2\), \((y-1)\) is positive. Since we are looking for increasing solutions at \(t=0\), we must have \((y - 1) \sin \pi t > 0\) at \(t=0\). Since \(\sin(\pi (0)) = 0\), there are no points at \(t=0\) where \((y - 1) \sin \pi t > 0\). Thus, no initial conditions \(y(0)=A\) lead to solutions that are increasing in time.

Key concepts

Graphing Utility

A graphing utility is an invaluable tool when it comes to understanding differential equations and their direction fields. This digital tool allows us to visualize how solutions to these equations behave under different circumstances. By inputing the differential equation \(y'(t) = (y - 1) \sin \pi t\) into a graphing utility, we can quickly generate a direction field that graphically represents the slope of solutions at various points in the plane defined by time \(t\) and function value \(y\).

These slopes, indicated by small lines or arrows, show the direction in which a solution curve would move if it passed through that point. The graph provides a snapshot of all possible solution trajectories, serving as a detailed map from which you can predict the behavior of solutions given particular initial conditions. As such, using a graphing utility simplifies complex numerical or algebraic methods and enhances our intuition about the system we're studying.

Constant Solutions

Constant solutions to differential equations are those where the value of the function does not change over time. Essentially, these are the steady states of our equation. In our exercise, finding the constant solutions involves setting the derivative equal to zero, \((y-1) \sin \pi t = 0\).

Identifying Constant Solutions

In this context, to find a constant solution, we focus on the \((y-1)\) term. This is because the value of \(y\) that makes \((y-1)=0\) will render our derivative zero for all values of \(t\), indicating that \(y=1\) is indeed a constant solution. This result corresponds to a horizontal line on the direction field, showing no change in \(y\) over time, regardless of the value of .

This understanding of constant solutions is particularly helpful if we're considering long-term behavior of a system modeled by the differential equation, as it informs us of the conditions under which the system will not evolve further.

Initial Conditions

Initial conditions, often denoted as \(y(0)=A\), play a crucial role in defining the specific solution to a differential equation out of a family of possible solutions. These conditions serve as a 'starting point' for the solution, dictating its behavior as time progresses.

Impact on Solution Trajectories

For the given differential equation, we're specially interested in distinguishing initial conditions leading to increasing solutions over time. The primary challenge here is the effect of the sine function, \(\sin \pi t\), which oscillates and changes sign. At \(t=0\), the sine function is zero, therefore, no positive value of \((y-1)\sin \pi t\) can be found, implying that no initial condition at this point leads to increasing solutions.

This conclusion is drawn by analyzing the derivative at \(t=0\); for an increasing solution, the derivative must be positive, but here it is zero. Thus, it is crucial to realize that the behavior of solutions is inherently tied to both initial conditions and the nature of the function encapsulated within the differential equation.