Slope Fields Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
For dy/dx=ydy/dx = y, compute slopes at (0,0)(0,0), (0,1)(0,1), (1,1)(1,1), (1,−1)(1,-1). Describe the solution family.

Solution

  1. 1
    Slope =y= y at each point.
  2. 2
    (0,0)(0,0): 0; (0,1)(0,1): 1; (1,1)(1,1): 1; (1,−1)(1,-1): −1-1.
  3. 3
    Solutions are y=Cexy = Ce^x: exponential curves.

Answer

Slopes: 0,1,1,−10, 1, 1, -1. Solutions: y=Cexy = Ce^x.
When slope depends only on yy (autonomous), the field has horizontal translation symmetry.

About Slope Fields

A graphical representation of a first-order DE dydx=f(x,y)\frac{dy}{dx} = f(x, y). At each point (x,y)(x, y) in the plane, draw a short line segment with slope f(x,y)f(x, y). The resulting pattern of segments shows the direction solutions must follow.

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More Slope Fields Examples

Example 2 medium

For [formula], find the zero-slope isocline and describe long-term behavior of solutions.

Example 3 easy

For [formula], compute slopes at [formula], [formula], [formula], [formula].

Example 4 medium

A slope field shows solutions curving toward the [formula]-axis from both sides. What can you infer

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