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
will be equal to 3.
- Since y=1 is an equilibrium point, the solution will be constant, in particular
will be equal to 1.
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