12–16. Sketching direction fields Use the window [-2, 2] x [-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(x) = sin y, y(−2) = 1/2

검증된 단계별 안내

Identify the differential equation given: \(\frac{dy}{dx} = \sin y\). This means the slope of the solution curve at any point \((x, y)\) depends only on the value of \(y\) at that point.

Set up the window for the direction field as \(x \in [-2, 2]\) and \(y \in [-2, 2]\). This defines the range of \(x\) and \(y\) values where you will plot the slope segments.

At various points \((x, y)\) in the window, calculate the slope \(\frac{dy}{dx} = \sin y\). Since the slope depends only on \(y\), for each horizontal line at a fixed \(y\), the slope will be constant.

Draw small line segments at each point with the slope \(\sin y\). For example, at \(y=0\), \(\sin 0 = 0\), so the slope is zero and the segments are horizontal. At \(y=\frac{\pi}{2}\) (approximately 1.57), \(\sin y = 1\), so the slope is 1, and so on.

To sketch the solution curve passing through the initial condition \(y(-2) = \frac{1}{2}\), start at the point \((-2, 0.5)\) on the graph and follow the direction indicated by the slope segments, moving from left to right, tracing a curve that aligns with the slopes.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Direction Fields

A direction field is a graphical representation of a first-order differential equation showing the slope of the solution curve at various points. Each small line segment indicates the slope y' = f(x, y) at that point, helping visualize the behavior of solutions without solving the equation explicitly.

Differential Equation y' = sin(y)

The given differential equation y' = sin(y) relates the slope of the solution curve to the sine of the dependent variable y. Understanding how sin(y) varies helps predict the slope patterns in the direction field, such as zero slopes at multiples of π and positive or negative slopes elsewhere.

Initial Condition and Solution Curves

An initial condition like y(−2) = 1/2 specifies a starting point on the direction field, allowing the sketching of a unique solution curve passing through that point. This curve follows the slope directions indicated by the field, illustrating how the solution evolves from the initial value.

12–16. Sketching direction fields Use the window [-2, 2] x [-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(x) = sin y, y(−2) = 1/2