Timeline for Orthogonal Trajectories Using Polar Coordinates. Correct Calculations, Two Different Answers?
Current License: CC BY-SA 4.0
13 events
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| Jan 26, 2023 at 16:55 | history | edited | Ted Shifrin | CC BY-SA 4.0 |
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| Jan 26, 2023 at 16:54 | comment | added | Ted Shifrin | @DavidK Yes, you’re quite right. I was sloppy at the very end. I’ll edit; thanks. | |
| Sep 26, 2019 at 11:19 | review | Suggested edits | |||
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| Mar 12, 2017 at 5:36 | vote | accept | The Pointer | ||
| Mar 10, 2017 at 19:50 | history | edited | Ted Shifrin | CC BY-SA 3.0 |
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| Mar 10, 2017 at 19:40 | comment | added | Rafa Budría | I tried to understand the curve in parametric form and taking the tangent vector. | |
| Mar 10, 2017 at 19:37 | comment | added | Ted Shifrin | @RafaBudría: Well, more than a cursory assertion is needed here. This is far from well-known. (I've taught calculus and multivariable calculus for over 40 years and certainly didn't know this off the top of my head.) | |
| Mar 10, 2017 at 19:36 | comment | added | Rafa Budría | In respect to your last edit: that was my point exactly. | |
| Mar 10, 2017 at 19:36 | comment | added | The Pointer | Thanks, Ted; I look forward to it. For further elaboration, the authors got $\dfrac{dy}{dx} = \dfrac{2xy}{x^2 - y^2}$ from $x^2 + y^2 = 2Cx$. They then said, "Unfortunately, the variables cannot be separated, so without additional techniques for solving differential equations we can go no further in this direction. However, if we use polar coordinates, the equation of the family can be written as $r = 2Ccos(\theta)$". And then they continue with their calculations, as stated above. | |
| Mar 10, 2017 at 19:35 | history | edited | Ted Shifrin | CC BY-SA 3.0 |
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| Mar 10, 2017 at 19:31 | comment | added | Ted Shifrin | Simmons is very astute, so I would trust him. I don't have the book to check. However, he is certainly not explaining what's going on here. Perhaps he is not claiming to have justified his solution. I'll add on a bit more. | |
| Mar 10, 2017 at 19:24 | comment | added | The Pointer | thanks for the elaborate response, Ted. This question was posed at the very beginning (chapter 1) of my differential equations book, so I assume that the author wanted to simplify it instead of assuming prior knowledge. Are you saying that both solutions (mine and the author's) are incorrect? The textbook is, "Differential Equations with Applications and Historical Notes, 3rd edition", by Simmons and Finlay. Can you please be more precise with regards to what's wrong with my/the textbooks calculations? I've only just begun studying differential equations so I really don't understand much. | |
| Mar 10, 2017 at 19:17 | history | answered | Ted Shifrin | CC BY-SA 3.0 |