Slope Fields Math Example 4

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

Example 4

medium
A slope field shows solutions curving toward the xx-axis from both sides. What can you infer about f(x,y)f(x,y) and long-term behavior?

Solution

  1. 1
    Above xx-axis (y>0y>0): slopes negative; below (y<0y<0): slopes positive.
  2. 2
    ff has opposite sign to yy: suggests f=−kyf = -ky, k>0k>0.
  3. 3
    Long-term: solutions converge to y=0y=0 (stable equilibrium).

Answer

f<0f < 0 when y>0y>0; f>0f > 0 when y<0y<0; y=0y=0 is a stable equilibrium.
Arrows pointing toward the axis indicate a stable equilibrium — characteristic of exponential decay.

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 1 easy

For [formula], compute slopes at [formula], [formula], [formula], [formula]. Describe the solution f

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].

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