Slope Fields: Answer to Example 2

Example: Consider the autonomous differential equation with the initial condition


where the graph of f(y) is given below

Draw the phase line for this differential equation, and classify the equilibrium points (critical points) as sinks, sources, or nodes.
Give a rough sketch of the slope field for this differential equation, and draw a few solutions into the slope field.
Consider the solution to the differential equation which satisfies the initial condition y(1)=2. Find


Same as in 3., if y(2)=1, that is find



Here is a picture of the phase line (with the slope field):

The equilibrium points are y=0, y=1 and y=3.

The equilibrium point y=0 is a source, y=1 is a node, and y=3 is a sink.

Below you can see the same picture with some solutions (in blue):

Since y(1)=2, the solution will increase over time and eventually approach the sink at y=3. Thus displaymath35 will be equal to 3.

Since y=1 is an equilibrium point, the solution will be constant, in particular displaymath35 will be equal to 1.

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Author: Helmut Knaust
Last Update 6-22-98

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