Question

Consider the following differential equations. A detailed direction field is not needed. a. Find the solutions that are constant, for all \(t \geq 0\) (the equilibrium solutions). b. In what regions are solutions increasing? Decreasing? c. Which initial conditions \(y(0)=\) A lead to solutions that are increasing in time? Decreasing? d. Sketch the direction field and verify that it is consistent with parts \((a)-(c)\). $$y^{\prime}(t)=\cos y, \text { for }|y| \leq \pi$$

Short answer

b. In which regions are the solutions of the given first-order differential equation \(y^{\prime}(t) = \cos y\) increasing or decreasing? c. What initial conditions lead to increasing or decreasing solutions of the given first-order differential equation \(y^{\prime}(t) = \cos y\)? d. Verify that a sketched direction field for the given first-order differential equation \(y^{\prime}(t) = \cos y\) is consistent with the findings in parts (a), (b), and (c).

Step-by-step solution

Find equilibrium solutions

To find equilibrium solutions, we are looking for solutions that remain constant for all \(t \geq 0\). For a solution to be constant, its derivative must equal zero, i.e., \(y^{\prime}(t) = 0\). From the given differential equation, we have: $$0 = \cos y$$ The equilibrium solutions are the values of y for which \(\cos(y) = 0\). These occur when \(y = \pm \frac{\pi}{2} + n\pi\), where \(n\) is an integer, and \(|y| \leq \pi\). There are only two such values that fall in this range: $$y_{1} = -\frac{\pi}{2}, \quad y_{2} = \frac{\pi}{2}$$ These are our equilibrium solutions.

Determine increasing or decreasing regions

To determine where solutions are increasing or decreasing, we need to analyze the sign of \(y^{\prime}(t) = \cos(y)\). If \(\cos(y) > 0\), the solution is increasing, and if \(\cos(y) < 0\), the solution is decreasing. We can use the values of the equilibrium solutions found in step 1 to split the range \(|y| \leq \pi\) into regions like this: 1. \(-\pi \le y < -\frac{\pi}{2}\) : Here, \(\cos(y) < 0\), so solutions are decreasing in this region. 2. \(-\frac{\pi}{2} < y < \frac{\pi}{2}\) : Here, \(\cos(y) > 0\), so solutions are increasing in this region. 3. \(\frac{\pi}{2} < y \le \pi\) : Here, \(\cos(y) < 0\), so solutions are decreasing in this region.

Determine initial conditions leading to increasing or decreasing solutions

We have already found the regions in step 2 where solutions are increasing and decreasing. Given an initial condition \(y(0) = A\), we can find whether the solution will be increasing or decreasing by checking which region the value of A falls in: 1. If \(-\pi \le A < -\frac{\pi}{2}\), then the solution is decreasing. 2. If \(-\frac{\pi}{2} < A < \frac{\pi}{2}\), then the solution is increasing. 3. If \(\frac{\pi}{2} < A \le \pi\), then the solution is decreasing.

Sketch the direction field and verify consistency with parts (a) - (c)

To sketch the direction field, we indicate the direction of the solutions at different points \((t, y)\) in the \(ty\)-plane based on the sign of \(y^{\prime}(t) = \cos(y)\): 1. In the ranges \(-\pi \le y < -\frac{\pi}{2}\) and \(\frac{\pi}{2} < y \le \pi\), the direction field will have negative slopes (decreasing solutions). 2. In the range \(-\frac{\pi}{2} < y < \frac{\pi}{2}\), the direction field will have positive slopes (increasing solutions). We can also sketch the horizontal lines representing the equilibrium solutions at \(y = -\frac{\pi}{2}\) and \(y = \frac{\pi}{2}\) where the slope is zero. Upon sketching the direction field, it should be easy to verify that it is consistent with our findings from steps 1, 2, and 3.

Key concepts

Equilibrium Solutions

In the study of differential equations, equilibrium solutions play a crucial role. They represent values where the solution remains constant over time. For a solution to be constant in our context, its derivative, denoted as \( y'(t) \), must be zero. For the given differential equation \( y'(t) = \cos y \), the goal is to find \( y \) values where \( \cos y = 0 \). This equation holds true where the cosine function equals zero. Specifically, these values are given by \( y = \pm \frac{\pi}{2} + n\pi \), where \( n \) is any integer. Within the set constraint \( |y| \leq \pi \), the only equilibrium solutions are \( y = \frac{\pi}{2} \) and \( y = -\frac{\pi}{2} \). These specific values are critical as they dictate points of constant solutions which serve as the baseline for analyzing changes in behavior around these points.

Direction Field

A direction field, or slope field, provides a visual representation of the solutions to a differential equation. It's created by sketching small line segments or arrows on a coordinate plane, indicating the slope of the solution at various points \((t, y)\). For the differential equation \( y'(t) = \cos(y) \), these segments depend solely on the value of \( y \) and not on \( t \), due to the function's autonomous nature. The direction field helps us visualize how solutions behave in different regions. At equilibrium solutions, given by \( y = \frac{\pi}{2} \) and \( y = -\frac{\pi}{2} \), the line segments are horizontal since the slope (\( y' \)) is zero. In other intervals, the direction field gives an upward slope where \( \cos(y) > 0 \) and a downward slope where \( \cos(y) < 0 \). These visual cues are helpful in predicting the behavior of solutions.

Increasing and Decreasing Solutions

The behavior of a solution as increasing or decreasing is determined by the sign of its derivative. When \( y'(t) = \cos(y) > 0 \), the function is increasing, meaning that its value grows as \( t \) increases. Conversely, if \( \cos(y) < 0 \), the solution is decreasing, and its value declines over time. For the interval \(-\pi \le y < -\frac{\pi}{2}\) and \(\frac{\pi}{2} < y \le \pi \), the cosine function is negative, leading to decreasing solutions. Meanwhile, solutions are increasing in the interval \(-\frac{\pi}{2} < y < \frac{\pi}{2} \), where \( \cos(y) \) is positive. Understanding these intervals is essential for identifying the behavior of solutions as they move away from equilibrium points and how they possibly return to these states over time.

Initial Conditions

Initial conditions specify the starting point of a solution \((t_0, y_0)\) in the context of differential equations. To determine the behavior of solutions for the problem \( y'(t) = \cos(y) \), initial conditions \( y(0) = A \) guide us on whether the solution is increasing or decreasing. If the initial condition \( A \) lies in the range \(-\frac{\pi}{2} < A < \frac{\pi}{2}\), the solution will be increasing because \( \cos(A) > 0 \). Conversely, if \( A \) is in the ranges \(-\pi \le A < -\frac{\pi}{2}\) or \(\frac{\pi}{2} < A \le \pi\), the solution is decreasing, since \( \cos(A) < 0 \). Initial conditions are crucial since they define the trajectory of the solution from the starting time \( t = 0 \) onwards, ultimately tailoring the path depending on where \( A \) lies relative to the intervals of increase and decrease.