03. Autonomous Equations

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Slope Field¶

Definition: Slope Field

A slope field is a tool we can use to learn information about the solutions to an ODE even when we cannot solve it completely.
Using ODE, we look at many points in the plane and calculate the slope of a tangent to any solution \(y(x)\) at that point. By doing this for many points, we can see what graphs of solutions look like.

Autonomous Equations¶

An ODE is autonomous if it is in the form \(y’=f(y)\).
For example,

• \(y'=−y^2\)
• \(y'=4𝑦(1−𝑦)\)
  What does the slope field of an autonomous equation look like?
  Since \(y’\) depends only on y, the slope of any solution is constant along any horizontal line.

Given the ODE \(y’=-y^2\), we can find the slopes for any given y:

Actual Solutions to the ODE¶

For the ODE \(y’=-y^2\):

The equilibrium solution is \(y=0\).

To solve:

Here, the uniformity of the slope along each horizontal line is very clear, because each solution \(y = \frac{1}{x + C}\) is a right/left translation of the graph \(y = \frac{1}{x}\).

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y = \frac{1}{x + C_1}
y = \frac{1}{x + C_2}
y = \frac{1}{x + C_3}
C_1=-1
C_2=0
C_3=1

Sketching Solutions of an Autonomous ODE¶

\(y’=f(y)\)
We’ll be using the equation \(y’=4y(1-y)\).

1. Sketch the equilibrium solutions \(y=k\) for each root \(k\) of \(f\)
   1. \(4y(1-y)=0\)
      • \(y=1\)
      • \(y=0\)
   2. 
2. Sketch (on a separate pair of axes) the graph of \(f(y)\) as a function of \(y\), or at least investigate where \(f\) is positive/negative, increasing/decreasing, and find the extreme points.
   1. Sketch the function as \(y=4x(1-x)\)
   2. Find the roots (we already did this in step one)
      • 1, 0
   3. Examine the expanded form \(y=-4x^2+4x\)
   4. Identify the extreme points
      1. 2^nd derivative is \(y=-8x+4 \implies -8(x-\frac12)\)
      2. Root of 2^nd derivative is \(x=\frac12\)
   5. Graph is a “frowning” parabola, so our extreme point is a maximum
   6. 
3. Examine the “signs” of \(f(y)\):
   • For arguments \(y\) where \(f(y)<0\implies y'>0\implies y(x)\) is decreasing
   • For arguments \(y\) where \(f(y)>0\implies y'>0\implies y(x)\) is increasing
   • For arguments \(y\) where \(f(y)=0\implies\) we already know this is an equilibrium solution
   • 
4. Find the concavity of the solutions
   • In an autonomous ODE, \(y’’ = f’\cdot f\)
   • \(f’\cdot f > 0\)
     • \(y(x)\) is concave up
       • \(f’, f > 0\) i.e. \(f(y)\) is positive and increasing
       • \(f’, f < 0\) i.e. \(f(y)\) is negative and decreasing
   • \(f’\cdot f < 0\)
     • \(y(x)\) is concave down
       • \(f’ > 0 f < 0\) i.e. \(f(y)\) is negative and increasing
       • \(f’ < 0, f > 0\) i.e. \(f(y)\) is positive and decreasing
   • \(f’=0\) and changing sign \(\iff\) \(y\) is an extreme point of \(f\)
     • y(x) has an inflection point
   • In our example, \(f’\) only changes signs at it’s single root (see step 2.4.2), so that is our single point of inflection.

Given the above calculations and sketches, we examine the behavior of f, f’, y’’ as y passes through the values \(0, \frac12,1\), since those were our equilibrium solutions and inflection points:

We can now sketch the slope field of the solutions:

Summary¶

Classification of Autonomous ODE Solutions¶

The equilibrium solutions of an AODE can be classified as one of three types:

1. Stable:
   • If \(y\) strays slightly from the equilibrium, then it will move towards it as \(x\) increases
2. Unstable:
   • If \(y\) strays slightly from the equilibrium, then it will move further away it as \(x\) increases
3. Semi-Stable:
   • Stable in one direction of \(y\) (up or down), and unstable in the other direction.