Orthogonal Trajectories and Differential Equations - Calculus 2

In this video, I will show you how to find the orthogonal trajectories using differential equations. This is a very important topic that students learn in Calculus 2. A differential equation is an equation with an unknown variable and its derivatives. Two trajectories are orthogonal to each other when their intersections form right angles. A simple example are two lines intersecting each other at a right angle, or a circle at the origin with a line passing through the origin and intersects the circle at a right angle. Orthogonal trajectories come up a lot in Physics classes. For example, in an electrostatic field, the lines of force are orthogonal to the lines of constant potential. Also, the streamlines in aerodynamics are orthogonal trajectories of the velocity-equipotential curves. To find the orthogonal slopes, simply take the negative reciprocal of the tangent equation of the family of curves. I go through many examples in the video. Usually, finding orthogonal trajectories will give you separable differential equations, which are easy to solve.

[@nomann5244]

This is the best,simple and intutive explation ever in a short time!

[@QuocDatPhung]

Thanks Nomann!! I'm glad you enjoyed my video! I would really appreciate if you could share with your classmates or kindly subscribe ~ you can find all of my Calculus II videos in this link: ua-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@aadhavanraju9944]

Excellent explaination. Before i watch, i was zero at this topic and you made me to love this topic, thank you .all the best ✨

[@QuocDatPhung]

So nice of you! It'd mean so much if you could subscribe! Also, you can find all of my Calculus videos here: ua-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@ashishsarker6139]

thank you for sharing your knowledge I get the basic idea now on how to sketck

[@QuocDatPhung]

You're very welcome Ashishsarker! If you really enjoy my videos, please kindly share with your classmates! Also, you can find the rest of my Calculus 2 videos in this link: ua-cam.com/video/lQr6hpVPbUg/v-deo.html

[@Omega.Animations]

i love the way you gave a practice question at the end. This makes sure one understands it..
But i have a question. After my integration of both sides, i got y^2 = - x^2 + c to become y^2 + x^2 = c but after reconsidering the way i distributed the negative sign, i got the soultion. (ie x^2 - y^2 = c). What's the best way to distribute the negative sign because it may lead to wrong answer if not done correctly?

[@QuocDatPhung]

Hi Omega! if you have y^2 = - x^2 + c you plus both sides with x^2 to get x^2 + y^2 = c. That should be the way to do it :)

[@Omega.Animations]

@@QuocDatPhung Thanks

[@QuocDatPhung]

@@Omega.Animations I'm glad you found my video helpful! You can find all of my Calculus 2 videos here: ua-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@chathurangajayathu9689]

Thanks❤️💚 great job

[@raghvendrayadav-e3c]

So simple and effective ❤❤

[@QuocDatPhung]

Thank you Raghvendrayadav! I'm really glad you like my explanation! If you know anyone who needs help with this class, please kindly share with them and also subscribe to support me (it means a lot) ~ you can find all of my Calculus 2 videos in this link: ua-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@ecemevakilinc7279]

Great explanation, thank you 😊

[@QuocDatPhung]

You're very welcome! If you like my Calculus videos, you can find all of them here in this playlist: ua-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@SmartStudiesAcademy1994]

Big fan👍

[@QuocDatPhung]

Thanks Smart Studies Academy! I'm really glad you like my explanation! If you know anyone who needs help with this class, please kindly share with them and also subscribe to support me (it means a lot) ~ you can find all of my Calculus 2 videos in this link: ua-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@Solemnn30]

Very good explanation 🎉Thank you ✨

[@QuocDatPhung]

You're welcome 😊! If you like my Calculus videos, you can find all of them in this playlist: ua-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@xaveragekid]

Thank you! Great explanation!!!

[@QuocDatPhung]

You're very welcome! You can find all of my Calculus II videos here: ua-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@keyb]

Super useful, but for the final problem I end up with
y^2 - x^2 = C (or) 1/2 y^2 = 1/2 x^2 + C
I am unsure as to how I am messing up, I believe it is somewhere with the negative sign distribution, but double checking all my work isn't getting me many results.
I think maybe I'm doing something wrong with the reciprocal or with the differentiation of k/x (which I believe is -k/x^2).
Any advice would be super useful!

[@QuocDatPhung]

That's also correct, because C is a constant. Suppose your equation is y^2 - x^2 = C multiply both sides of the equation by -1.
You get -y^2 + x^2 = -C or x^2 - y^2 = -C
However, because -c is still a constant, you can say let "constant"=-c
You end up with x^2 - y^2 = constant
Bingo that's the correct answer! Let me know if that makes sense!