Differential Equations: Orthogonal Trajectories: Example 1

[@mes]

I don't always determine the orthogonal trajectories of a family of curves but when I do I usually set up a differential equation and look at the negative reciprocal of the slope to form a second differential equation ;) View Video Notes on Steemit: steemit.com/mathematics/@mes/differential-equations-orthogonal-trajectories-example-1

[@durga18991]

For complete notes of this chapter visitkzbin.info/aero/PLN1owt4VsNz-nolZIF-Cjjfs8fqi4Xnkp

[@JacksonBucholz]

Hey quick question! Is it possible to solve these problems without setting up implicit differential equations? For instance, in the first example can we just solve for y... y = sqrt(x/k) y' = 1/2sqrt(xk) y' (perpendicular) = -2sqrt(x*k)..... and then solve < that line for our function? Also, can you clarify please why we have to get rid of k. What do you mean by "simultaneously valid for all values of k"? Thanks so much!

[@mes]

Thanks for asking. First off all your derivative is missing a 1/k term (by the chain rule). so y' = 1/(2k sqrt(x/k)). You can solve it directly as you are doing but I believe you will be stuck with the k terms. I got rid of k so that we don't need to worry about it for the derivatives of the orthogonal trajectories. If you include k then you would need to include it for every derivative. If you get rid of it, then they are "simultaneously" removed for all derivatives.

[@jasmines7354]

Thankyou! This helped me a-lot with understanding the concept :D

[@ILoveTinfoilHats]

Why is the intro keemstar

[@has9629]

Thank you!

[@KanoBoom]

Odd thing is in my textbook the solution is 2x^2+y^2=c^2 I’m not sure why

[@mes]

That is just a different representation of the same formula I got in the video. We can rearrange my answer to get yours: x^2 + (y^2)/2 = C --> multiply by 2 on both sides: 2*x^2 +y^2 = 2C. 2C is just a constant (so is c^2 just a constant) so your textbook wrote it in the typical ellipse form by letting c^2 = 2C.

[@KanoBoom]

Math Easy Solutions Elaborate on typical ellipses form? ( I thought Wikipedia ellipse could explain it more but I realised I can’t seem to find one)

[@itsSHKR]

i believe its "or-thawgonal" .... not "ortha-gone-al".... :3

[@wintergreen2993]

very hard to listen to the vid purely off the pronunciation

[@sheabryan5354]

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[@needheartranken]

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[@HoboW1thShot9un]

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[@McZeoed]

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[@Mittenssss]

Lol

[@mes]

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