JSFH Math 221h, Fall 2008, Quiz 2 Key
Name: J. Smyth
Investigation A, p. 31 of text: We start with a slople field and typical solution curves for the differential equation 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 , in an enlarged range of 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 , 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 . Look for straight line solutions. First load the DEtools package:
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEld2l0aEYnLyUlc2l6ZUdRIzExRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiYtRiw2JlEoREV0b29sc0YnRi9GMkY1Ri8vJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GNlEnbm9ybWFsRidGL0ZDLUkjbW9HRiQ2LlEiOkYnRi9GQy8lJmZlbmNlR0ZCLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRllGL0ZARkM=
print(
Here is our differential equation defined as a variable:
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print(
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print(
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print(
Next, we create a graph with selected solutions:
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Comment: From the direction field, it does appear that there might be straight line solutions, but we didn't generate one explicitly. However, after doing part (a), we added the last three initial conditions to verify that we did get straight line solutions.
(a) Substitute 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 in the differential equation to determine what values of LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEiYUYnLyUlc2l6ZUdRIzExRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiw2JlEiYkYnLyUlc2l6ZUdRIzExRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRitGMi9GOVEnbm9ybWFsRic= must be in order to get a solution.
Solution: We do this by hand: 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 , so the equation becomes 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 . But the left-hand side is constant, and the right-hand side is never constant unless LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiYUYnLyUlc2l6ZUdRIzExRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSIxRidGL0Y8Ri9GPA== , in which case 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 , so that 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 etc. So the straight line solutions are of the form 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 . We'll add these to our graph above.
print(
(b) Next, let's find the general solution using Maple:
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
LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=
print(
Ly1JInlHNiI2I0kieEdGJSwmRiciIiIqJiIiI0YpLUknYXJjdGFuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJiwoRidGKUYrISIiSSRfQzFHRiVGNEYpLCZGJ0YpRjVGNEY0RilGNA==
Now for some plotting. Add various initial conditions to get some solutions
%;
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
print(
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
LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=
%;
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Comment: No, it doesn't seem like any particular choice of the constant C will lead to a straight line solution, although large values like LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkX0MxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEkMTAwRidGOUY5 look linear in this range. Looking at the formula for a general solution, we see why. At some point in our graph range (except the large choice), the denominator of the fraction in the arctan vanishes and the argument becomes infinite. In any case, it is never a linear term, so this formula couldn't possibly account for linear solutions. As we learned later, these are so-called singular solutions.
Note: To make this assignment fit 2 printed pages, I selected all but the title and downsized the font to 11 pt, and then downsized the graphs a bit by grabbing and moving a corner.
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