17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.


c. Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing?


y'(t) = cos y for |y| ≤ π

Verified step by step guidance

1. First, understand that the differential equation is given by \(y'(t) = \cos y\), where \(y\) depends on \(t\) and \(|y| \leq \pi\).
2. Recall that the sign of \(y'(t)\) determines whether the solution \(y(t)\) is increasing or decreasing at time \(t\). Specifically, if \(y'(t) > 0\), then \(y(t)\) is increasing; if \(y'(t) < 0\), then \(y(t)\) is decreasing.
3. Since \(y'(t) = \cos y\), analyze the values of \(\cos y\) for \(y\) in the interval \([-\pi, \pi]\). Identify where \(\cos y\) is positive, zero, or negative.
4. Determine the initial conditions \(y(0) = A\) such that \(\cos A > 0\) (leading to increasing solutions) and \(\cos A < 0\) (leading to decreasing solutions). Also note the points where \(\cos A = 0\), which correspond to equilibrium solutions where \(y(t)\) does not change.
5. Summarize the intervals of \(A\) where solutions increase or decrease based on the sign of \(\cos A\), considering the periodic and even nature of the cosine function within the given domain.

Key Concepts

Differential Equations and Initial Conditions

A differential equation relates a function and its derivatives. The initial condition y(0) = A specifies the starting value of the solution, which influences the behavior of the solution curve over time. Understanding how initial values affect solutions is key to predicting whether the solution increases or decreases.

Sign of the Derivative and Monotonicity

The sign of y'(t) determines whether the solution y(t) is increasing or decreasing. If y'(t) > 0, the solution is increasing; if y'(t) < 0, it is decreasing. Analyzing the expression y'(t) = cos y helps identify intervals of y where the solution grows or shrinks.

Properties of the Cosine Function on the Interval |y| ≤ π

Cosine varies between -1 and 1 and is positive on (-π/2, π/2) and negative on (π/2, π) and (-π, -π/2). Since y'(t) = cos y, the sign of cos y depends on y's value, which determines whether the solution increases or decreases. Recognizing these intervals is essential for classifying initial conditions.