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.

[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!

[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?

[ashishsarker]

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

[aadhavanraju]

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

[nomann]

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