**Example:** Consider the autonomous differential equation with the initial condition

where the graph of *f*(*y*) is given below

**1.**- Draw the phase line for this differential equation, and classify the equilibrium points (critical points) as sinks, sources, or nodes.
**2.**- Give a rough sketch of the slope field for this differential equation, and draw a few solutions into the slope field.
**3.**- Consider the solution to the differential equation which satisfies the initial condition
*y*(1)=2. Find **4.**- Same as in 3., if
*y*(2)=1, that is find

**Solution:**

- 1.
- 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. - 2.
- Below you can see the same picture with some solutions (in blue):
- 3.
- Since y(1)=2, the solution will increase over time and eventually approach the sink at
*y*=3. Thus will be equal to 3. - 3.
- Since y=1 is an equilibrium point, the solution will be constant, in particular
will be equal to 1.
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*Author: Helmut Knaust*Last Update 6-22-98

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