Problem 2 - Calculus Early Transcendentals

Question

Sketch the slope field for \(y^{\prime}+y=2\) at the 25 gridpoints \((x, y),\) where \(x=0,1, \ldots, 4\) and \(y=0,1, \ldots, 4\)

Step-by-Step Solution

Answer

Calculate slopes using \( y' = 2 - y \) and sketch lines at grid points accordingly.

Step 1: Understand the Differential Equation

The given differential equation is \( y' + y = 2 \). Our goal is to sketch the slope field for this equation by determining the slope of the solution curve at each grid point \((x, y)\).

Step 2: Express the Equation in Standard Slope Form

Solve the equation \( y' + y = 2 \) for \( y' \). Subtract \( y \) from both sides to obtain \( y' = 2 - y \). This expression gives us the slope of the solution curve at any point \((x, y)\).

Step 3: Calculate Slopes at Grid Points

Calculate the slope \( y' = 2 - y \) for each grid point \((x, y)\) where \( x = 0, 1, 2, 3, 4 \) and \( y = 0, 1, 2, 3, 4 \). For instance, at \((x, y) = (0,0)\), \( y' = 2\); at \((x, y) = (0,1)\), \( y' = 1\); at \((x, y) = (0,2)\), \( y' = 0\); and so on.

Step 4: Sketch the Slope Field

At each point \((x, y)\), draw a small line segment with the corresponding slope calculated in the previous step. For example, at \( (0,0) \), draw a line with a slope of 2, which is steeper compared to that at \((0,1)\) with a slope of 1.

Step 5: Analyze Slope Field for Patterns

Observe the general direction and pattern of the solution curves suggested by the slope field. Slope segments will show that as \( y \) increases, the slopes \( y' = 2 - y \) decrease, eventually becoming negative as \( y > 2 \).

Key Concepts

Slope Fields

One of the fascinating aspects of differential equations is the concept of slope fields. Understanding slope fields is like peering into the visual representation of differential equations. A slope field gives us a graphical view of all the potential solution curves for a given differential equation without actually solving it analytically.
To create a slope field, you calculate the slope of the differential equation at a series of points, or grid points. In our case, from the equation \(y' = 2 - y\), you observe how the slope changes with different values of \(y\).

• The slope is steepest where the calculated value is highest, indicating a rapid change in the solution curve.
• These slopes are represented by short line segments at grid points, which together form the slope field.

The collection of these line segments provides insight into the behavior of possible solution curves, forming a kind of "direction map" for the equation's solutions.

Solution Curves

Solution curves are the trajectories that actual solutions of the differential equation will take. In the context of slope fields, these curves fit seamlessly into the "direction map" created by slope segments. Each solution curve follows the path of steepest ascent or descent dictated by these slopes, threading through the field in harmony.
The differential equation \(y' = 2 - y\) offers a wide range of solutions, but they all share a common feature defined by the slope field. As you consider solution curves:

• Note how passing through each grid point, the solution curve adjusts to match the slope of the field.
• The guided path helps predict where the curve will lead, illustrating the general behavior and long-term trends of solutions.
• The set of solution curves forms what's known as a "family of solutions"—each curve differing slightly based on initial conditions.

This visual feedback is crucial for understanding the dynamics of differential equations, sometimes more tangibly than algebraic expressions alone.

Grid Points

Grid points serve as the foundation for constructing slope fields. They are predetermined coordinates where you evaluate the differential equation to determine the slope at those particular locations. For our exercise, grid points are set at \((x, y)\) values, where both \(x\) and \(y\) range from 0 to 4.
At each grid point, the specific step is to:

• Insert the \(y\) value into \(y' = 2 - y\) to find the slope at that point.
• Draw a line segment that represents this slope, contributing to the overall slope field.

This preparatory step is crucial because without these calculated slopes, there would be no basis for forming solution curves visually. The distributed grid points allow for a more complete and accurate representation of possible solution behaviors across a specified region.