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USDescartes @USDescartes
Joined January 2009
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USDescartes
@USDescartes
12 Nov 2022
#POTD #Math #Geometry #Advanced Problem of the Day! This is possibly on the more difficult side, but this one came to me as I was falling asleep last night, so I thought I'd write it up. Enjoy! And don't forget to explain your reasoning!
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Problems of the Weekend! I thought I'd switch it up this weekend to present something more accessible. Take your time and plan carefully! #POTD #Math #Algebra #Medium
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Problem of the Day! Question 10 in the line of definite integral equations. Not many more left, as I want to move on to something different. #POTD #Math #Calculus #DEs #IEs #DIEs #Medium
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Problem of the Day! Question 9 in the sequence of definite integral equations. Just a few more of these. This one is a little tricky. #POTD #Math #Calculus #DEs #IEs #DIEs #Medium
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With one day (3 games) to go in the round of 32, the only game I've missed is the Germany V Paraguay game.
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Problem of the Day! Question 8 in the line of definite integral equations, particularly for those who might want an easier one to get started with. #POTD #Math #Calculus #DEs #IEs #DIEs #Easy
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Problem of the Day! Question 7 in the line of definite integral equations, particularly for those who might want an easier one to get started with. #POTD #Math #Calculus #DEs #IEs #DIEs #Easy
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Problem of the Day! Question 6 in the line of definite integral equations. #POTD #Math #Calculus #DEs #IEs #DIEs
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Problem of the Day! Here's Question 4 in my sequence of integral equations... and I THINK the first really interesting one. #POTD #Math #Calculus #DEs #IEs #MediumDifficulty
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Nice work by @_Manuel_Ruiz_ and @Setu2352000. Given the differentiation and DE solving involved, though, it's like squaring both sides of an equation – we MUST check that our solution is valid.
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Problem of the Day! Another in the line of definite integral equations. #POTD #Math #Calculus #DEs #IEs #DIEs
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I figured out why these don't "work" sometimes as I am developing them, but I'm not sure I know how to predict when it will happen. For instance, in the ones I've written so far (except for Question 1), I(x) is odd. However, in solving the eventual DE, it doesn't result in an odd function. The answer, in those cases, is that I(x)≡0. I can give some examples later next week.
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Problem of the Day! This is the next problem in my sequence of integral equations. If people still need a hint for the weekend's Question 2, I'll publish one later. In the meantime, enjoy! #POTD #Math #Calculus #IEs #DEs #Easy
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Here's my solution to Question 3. I'll post Question 4 soon.
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In working these integral equations, I ran into something I had never really thought of before – at least not seriously: what does g'(-x) mean? There are two obvious possibilities.
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The way Mathematica treats it, it is the first definition: first you take the derivative of the function, then you make the plug in -x for whatever variable you are using for the function. I THINK this makes the most sense, but it sure puzzled me when I was noticing that g'[-x] = -D[g[-x],x]
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Nice, @_Manuel_Ruiz_ ! That is indeed what I got. I'll post my solution (method), which is different from yours, later today.
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And don't worry – they get more interesting from here!🙂
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Unlike the most self-referential integral I posted last week, where the internal integral didn't really mean anything, I've had fun over the last several days developing lots of integral equations. I'm finding them fascinating! Here are the first two easy ones:
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Here's my solution to Question 2:
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The more I've been thinking about these, by the way, the more I wonder who may have studied them or used them before. Oh, there are LOADS of places where I've seen integral equations that are the same as differential equations (eg, f = f' sin x ∫f ∫∫f), but I'm not sure how many functions equal to a definite integral I've ever seen. I mean, they make total sense – certainly in terms of, say, RLC circuits... but I seem to remember working those problems as indefinite integrals.
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