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Examples--First
Order Separable Equations
PowerPoint
Presentation of Solutions to Separable First Order Differential Equations
| Example 1: Solving a
Separable Differential Equation
In addition to the graphs shown at the
right you can look at various solutions corresponding to an initial
condition of the form y(0) = c by following this link to a DPGraph
of a surface and the plane y = c. The curve of intersection of
the surface and the plane when c = 1 is the graph of the solution to the
initial value problem above. You can use the scrollbar and activate
c to look at solutions for various values of c. You can also use the
z-slice feature.
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 The graph of the solution is shown above. Click on the graph
at the left to see an animation of the direction field
vectors moving across the screen for increasing values of x along with an
animated solution point. The lower graph shows the solutions for
y(0) = -1, -0.4, 0.4, and 1 along with the direction field. Click on
the graph to see an enlargement. First
Order DE Solution Grapher 
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| Example 2: Solving a Separable DE--Circles
yo = 1 yo
= 2 yo = 3
yo = 4 |
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| Example 3: Solving a Separable
DE--Logistics
Equation
Above is the graph of the
solution to the logistics equation.
QT animation
as y(0) varies from 0 to 40.
dy/dx = .01y(100-y), y(0) = 10.
The solution is y = 100ex / (9 + ex).
Here is an
Animation (Quicktime
version) of the changing graph of the solution as y(0) varies from
-50 to 150 with x between 0 and 10 and here is an Animation
(Quicktime version) with x between -10 and 10. Notice the significance of
the blue horizontal lines and their relationship to the zeroes of
.01y(100-y). Look at examples 3 and 4 in Section 2.1 discussing
autonomous first order differential equations (DE's of the form F(y,y') =
0 or in normal form dy/dx = f(y)). If the DE was modeling a
population then y(0) would have to be positive. If y(0) = yo
then
y = 100 is a singular solution (see pages 7-8 in
your text). Extra Credit
Problem--Link |
Solution
Details
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Click
here to see the solution graphed using Winplot. You may need
to download the file to your desktop and then use the freeware
Winplot to open the file (by opening Winplot, clicking on Window,
clicking on 2-dim, clicking on File, clicking on Open, and then opening
DEorder1Ex3. You can use the slider to vary the value of B (yo) from -50
to 150.
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Example
4: Section 2.2#36 (It is separable)
Note: y = 0.9 is a singular solution that
could not be obtained from the general solution shown above. Click
here to see animated solution graphs as yo varies from -2.1
to 3.9 demonstrating the "missing" singular solution. Quicktime
version
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Solution graphs for yo = 1
Click
here to see animated solution graphs as yo varies from -3
to 3. Quicktime version |
Example
5: Another variables separable example
The
picture on the right shows the graphs of particular solutions with y0
= 1/4 (blue) and y0
= -1/4 (red).
Click here or on the picture to
see an animation of solutions as y0 varies from -3 to 3.
In the case of the blue solution on the right, does y continue to increase
as x continues to increase beyond 8? Click
here for a pictorial answer.
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Quicktime
Version
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return
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