91Ó°ÊÓ

A slope field is a graphical representation of all possible solutions to a differential equation. For the equation \( y' = -y \), we examine how the rate of change of \( y \) depends solely on the value of \( y \). In a slope field:

• We draw small line segments at various points in the coordinate plane.
• Each line segment has a slope equal to \( -y \) for the corresponding \( y \) value of that point.

This forms a pattern showing how solutions curve over a range of initial conditions. For instance, at a point where \( y = 1 \), the slope is \(-1\), meaning the line segment there will tilt downwards. Conversely, if \( y = -1 \), the slope becomes \(1\), creating a small upward-tilting segment. By plotting these segments across many points, you produce a visual picture aiding the understanding of the behavior of solutions that follow the differential equation.

Separation of variables is an effective technique to solve differential equations where variables can be separated on different sides of the equation. Given the differential equation \( y' = -y \), we aim to isolate \( y \) and \( x \) to either side:

• Rearrange the equation to \( \frac{dy}{y} = -dx \).
• Integrate both sides separately. The left side becomes \( \int \frac{dy}{y} = \ln |y| \), and the right becomes \(-\int dx = -x + C \).

This separation effectively decomposes the problem into integrable parts. Integrating allows us to solve for \( y \) in terms of \( x \), which eventually helps determine a general solution. After integration, exponentiating both sides consolidates constants, leading to the expression \( y = Ce^{-x} \), encapsulating the family of all possible solutions.

The initial condition provides a specific point that the solution must pass through, ensuring that we select the correct one from all possible solutions. For \( y' = -y \) with the initial condition \( y(0) = 4 \), it means:

• The solution curve must intersect the \( y \)-axis at \( y = 4 \) when \( x = 0 \).
• This single point determines the constant \( C \) in our general solution \( y = Ce^{-x} \).

By substituting, we have \( 4 = Ce^0 \) or simply \( C = 4 \). The initial condition anchors the curve within the slope field and distinguishes the particular solution \( y = 4e^{-x} \). It ensures that the theory matches real-world conditions or specific scenarios described by the problem.

A particular solution is a specific instance of the general solution selected by applying initial conditions. With the general solution \( y = Ce^{-x} \) and initial condition \( y(0) = 4 \), we determined the particular solution as \( y = 4e^{-x} \).

• This solution uniquely follows the path through the slope field dictated by \( y' = -y \).
• It passes through \( (0, 4) \), which aligns with the given initial condition.
• Representing a real scenario, it illustrates how one particular path fits the broader family of solutions the equation describes, making it unique.

Plotting this on the slope field corroborates that the derived equation captures the behavior specified by the differential equation, rooted in the particular scenario imposed.