Integrating differential equations that have ln

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Discussion Overview

The discussion revolves around the integration of differential equations that involve natural logarithms, specifically focusing on the implications of rewriting solutions and the selection of particular solutions based on initial conditions. Participants explore the nuances of general versus particular solutions and the graphical representation of these solutions.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the validity of using the general solution to find particular solutions, suggesting that the original equation's (plus/minus) form should be used to capture both portions of the graph.
  • Another participant asserts that the lower portion of the graph can be obtained from the general solution by selecting C<0.
  • A follow-up inquiry raises the concern that choosing only C>0 would omit the lower portion of the graph, questioning the completeness of the solution when not considering both branches.
  • Some participants emphasize the necessity of choosing one branch to maintain the definition of a function, as a function must yield a unique value for each input.

Areas of Agreement / Disagreement

Participants express differing views on the implications of using the general solution versus the original (plus/minus) form. There is no consensus on whether the omission of the lower portion of the graph when using the general solution is a significant issue, and the discussion remains unresolved regarding the treatment of absolute values in the context of integration.

Contextual Notes

Participants note that the lack of absolute values in some problems may lead to confusion about the completeness of the solution, particularly in relation to the graphical representation of the function.

ecoo
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Hey guys, I have a question concerning the rewriting of a differential equation solution.

seperation of variables.png


In the example above, they rewrite [y=(plus/minus)e^c*sqrt(x^2+4)] as [y=C*sqrt(x^2+4)]. I understand that the general solution we get as a result represents all the possible functions, but if we were to attempt to find a particular solution given an initial condition (a point on the graph of the equation), I would think that we would plug in the coordinate into not the general solution but the one above it with the (plus/minus). The problem is that in some of the problems that is not the case - you plug in the coordinate into the general solution to find your equation.

The reason I would plug in the coordinate into the (plus/minus) equation is because that is the original equation contains above the x-axis values and below the x-axis values (when you graph the resulting equation). But if you plug a coordinate into the general solution, you only get the top portion of bottom portion of the graph depending on if y were less than or greater than 0 (because of the absolute value symbol). So plugging into the general solution kind of makes you lose a portion of the equation's graph.

Hopefully you guys can help clear up this confusion for me. If you need any clarification, please ask.

Thanks!
 
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One obtains the lower portion of the graph from the general solution (last line of your post) by choosing C<0.
 
andrewkirk said:
One obtains the lower portion of the graph from the general solution (last line of your post) by choosing C<0.

Does that mean we would be missing the bottom portion of the graph if we only choose C > 0. To see the whole equation, we'd have to choose C and -C. If we only chose C, shouldn't we mention that we are only seeing the y > 0 portion of the graph?

I guess my confusion comes from problems like this, where they don't put the y in absolute value (I don't know why), which you are supposed to do when integrating a ln derivative. By not putting the y in derivatives, aren't they losing the bottom portion of the graph? And if they did put the y in absolute values, when solving for the particular solution given a coordinate, wouldn't you have to use the (plus/minus) equation to get the top and bottom?
problem answer 1.png
 
You have to choose one branch or the other, or else y is not a function. A function of x must have a unique value for each value of x.
 
andrewkirk said:
You have to choose one branch or the other, or else y is not a function. A function of x must have a unique value for each value of x.

Thank you! Succinct and insightful :)
 

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