Slope Fields and Differential Equations

Key Concepts

• Definition of Slope
  • Slope of 0: Horizontal line
  • Slope of 1: Line at 45-degree angle
  • Slope of 2: Steeper line
  • Negative Slopes:
    • -1: Downward at 45 degrees
    • -2: Steeper downward line
  • Very Large Slopes: Approaching vertical line
  • Undefined Slope: Vertical line

Practice Problem: Differential Equation y' = x

1. Create a table:

   • Columns: x, y, y'
   • y' (slope) only depends on x:
     • When x = 0, y' = 0
     • When x = 1, y' = 1
     • When x = 2, y' = 2
     • When x = -1, y' = -1
     • When x = -2, y' = -2

2. Plotting the slope field:

   • At (0, y) for any y, the slope is 0 (horizontal line).
   • At (1, y) for any y, the slope is 1 (45-degree line).
   • At (2, y) for any y, the slope is 2 (steeper line).
   • Negative values follow similar patterns.

3. General Solution:

   • Integrate to find y:
   • y = (1/2)x² + C (parabolic shape).

Example: Differential Equation y' = y

1. Create a table for y':

   • When y = 0, y' = 0 (horizontal).
   • When y = 1, y' = 1 (slope = 1).
   • When y = 2, y' = 2 (slope = 2).
   • Negative values follow similar patterns.

2. Plotting the slope field:

   • Shows exponential growth.

3. General Solution:

   • y = Ce^x (exponential function).

Example: Differential Equation y' = x - y

1. Create a slope table:

   • (0,0) slope = 0
   • (1,1) slope = 0
   • Patterns for other coordinates:
   • Identify values for (1,0), (2,1), (3,2) (slope = 1).

2. Plotting the slope field:

   • Negative slopes increase in negative direction as y increases.
   • Expect nearly vertical lines for large values.

3. General Solution:

   • y = x - 1 (linear equation).

Matching Slope Fields to Equations

1. Pick a point from slope field to test equations.
2. Example at (0,1):
   • Check values for potential equations (eliminate wrong choices).
3. Correct Answer: Match to the correct differential equation.

Example: Circle Shape in Slope Fields

• Equation: y' = -x/y
• Integrate:
  • y dy = -x dx
  • y²/2 = -x²/2 + C
• Rearranging gives:
  • x² + y² = 2C (Equation of a circle).
• Conclusion: Circle equations can represent solution shapes.

Conclusion

• Slope fields provide insights into the shape of solutions for differential equations.
• Understanding slopes and key patterns is essential for creating and interpreting slope fields.