91Ó°ÊÓ

Differential equations are equations that relate a function with its derivatives. They describe how a quantity changes in relation to another. In the given exercise, we have the differential equation \( y' = y \), which is a simple first-order linear differential equation. This equation tells us that the rate of change of \( y \) with respect to \( x \) is equal to the current value of \( y \).
Understanding such equations is important as they appear in various scientific fields to model real-world phenomena, such as population growth, heat conduction, or motion. To solve differential equations like this one, we find a function \( y(x) \) that satisfies the given relationship throughout its domain. This often involves mathematical techniques like integration or special transformations.

A solution curve of a differential equation is a curve in the coordinate plane that represents a set of solutions to the differential equation for certain initial conditions. For \( y' = y \), a general solution is found as \( y = Ce^x \), where \( C \) is a constant.
The special points given in the exercise - \((0,1)\), \((0,2)\), and \((0,-1)\) - dictate the particular solutions for each case. These specific requirements help us determine the exact value of \( C \) for each initial condition:

• Through \((0,1)\), the curve is \( y = e^x \)
• Through \((0,2)\), it's \( y = 2e^x \)
• Through \((0,-1)\), it becomes \( y = -e^x \)

These curves represent exponential growth or decay and have specific characteristics based on their respective initial conditions.

Slope analysis is a method used to understand how the solution of a differential equation behaves at various points in the plane. It involves examining the slopes given by the differential equation at those points. For \( y' = y \), the slope at any point \((x,y)\) is exactly the value of \( y \) at that point.
This means that whenever \( y \) is positive, the slope will be positive, indicating upward growth. Conversely, if \( y \) is negative, the slope will be downward, suggesting exponential decay. With a slope of zero, there's neither growth nor decay. These slopes can be visualized through a slope field, where short line segments represent different slopes at various points, assisting us in piecing together a general view of solution behaviors.

A slope field is a graphical tool used to represent the behavior of a differential equation visually. It's a grid of lines overlaid on a plane, where each line indicates the slope of the solution curve at that point. By observing these tiny line segments, we can infer the shape and direction of potential solution curves even before solving the equation analytically.
For the equation \( y' = y \), creating a slope field involves drawing lines with slopes equal to the \( y \)-values at selected points. Then, you can add the solution curves that align with initial given points like \((0,1)\), \((0,2)\), and \((0,-1)\).

• At \((0,1)\), draw the curve \( y = e^x \)
• At \((0,2)\), illustrate \( y = 2e^x \)
• At \((0,-1)\), show \( y = -e^x \)

These curves depict specific trajectories on the field, showing how the solutions change across the plane, encapsulating both exponential growth and decay.