AHL 5.15—Slope fields Notes

Slope Fields in Differential Equations

Concept of Slope Fields

Slope fields, also known as direction fields, are graphical tools used to visualize solutions to first-order differential equations without solving them explicitly.

• A slope field consists of short line segments plotted at various points in the xy-plane.
• Each line segment represents the slope of the solution curve passing through that point.
• The slope at each point is determined by the differential equation.

For a differential equation of the form:

$$ \frac{dy}{dx} = f(x,y) $$

The slope field shows the value of $f(x,y)$ at various points $(x,y)$ in the plane.

Interpreting Slope Field Diagrams

To interpret a slope field diagram:

1. Observe the overall pattern of the slopes
2. Look for regions where slopes are similar or change dramatically
3. Identify any horizontal or vertical lines (where slopes are zero or undefined)
4. Notice any symmetry or periodicity in the field.

Example Slope Field Diagram

Consider the differential equation $\frac{dy}{dx}=x−y$

In this slope field:

• Slopes are positive in the region where x > y
• Slopes are negative where x < y
• Slopes are zero along the line y = x
• As y increases for a fixed x, slopes become more negative
• As x increases for a fixed y, slopes become more positive

Using Slope Fields to Visualize Solutions

Slope fields allow us to sketch approximate solution curves without solving the differential equation. To sketch a solution:

1. Choose a starting point
2. Draw a short line segment following the slope indicated at that point
3. Move to the end of that segment and repeat the process
4. Continue this process to trace out the solution curve

Sketching a Solution Curve Through a Given Point

To approximate a solution curve, follow these steps:

1. Identify the starting point: Suppose we want to sketch the solution passing through $(0,1)$.
2. Observe the slope at that point: From $\frac{dy}{dx} = x - y$, we calculate: $$\frac{dy}{dx} = 0 - 1 = -1$$So, the curve at $(0,1)$ should initially slope downward.
3. Move step-by-step in small increments:
   1. Follow the slope direction from each nearby point.
   2. The curve gradually adjusts to the local slopes, forming an approximation of the actual solution.

Analyzing Slope Fields to Understand Solution Behavior

Slope fields provide valuable insights into the behavior of solutions:

1. Equilibrium Solutions: Horizontal line segments in the slope field indicate constant solutions (where $\frac{dy}{dx}=0$).
2. Stability:
   1. If nearby slopes point towards an equilibrium solution, it's stable.
   2. If nearby slopes point away, it's unstable.
3. Asymptotic Behavior: Observe where solution curves tend to as x approaches infinity.
4. Periodicity: Look for repeating patterns in the slope field that might indicate periodic solutions.
5. Uniqueness: If slopes are well-defined everywhere, each point has a unique solution passing through it.

Mathematical Foundations

The theoretical basis for slope fields comes from the fundamental theorem of existence and uniqueness for first-order ODEs. For an equation $ \frac{dy}{dx} = f(x,y) $, if f is continuous and satisfies a Lipschitz condition, then:

1. A solution exists for any initial condition $(x_0,y_0)$
2. This solution is unique
3. The solution depends continuously on the initial condition

This theorem justifies our use of slope fields to visualize and analyze solutions.