Slope Fields Math Example 2

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

Example 2

medium
For dy/dx=x−ydy/dx = x-y, find the zero-slope isocline and describe long-term behavior of solutions.

Solution

  1. 1
    Zero-slope isocline: x−y=0x-y=0, i.e., y=xy=x.
  2. 2
    Above y=xy=x: slope negative (solutions decrease toward y=xy=x).
  3. 3
    Below y=xy=x: slope positive (solutions increase toward y=xy=x).
  4. 4
    General solution y=x−1+Ce−xy = x-1+Ce^{-x}: approaches y=x−1y=x-1 as x→∞x\to\infty.

Answer

Zero isocline: y=xy=x; solutions approach y=x−1y=x-1 as x→∞x\to\infty.
Isoclines organize the slope field. Solutions are attracted toward the line y=x−1y=x-1.

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 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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