Differential Equations: Orthogonal Trajectories: Example 1

In this video I go over a recap on orthogonal trajectories as well as an example on how to go about solving for a family of orthogonal trajectories to the parabolas x = k*y^2, where k is a constant. The first step is to write the parabolas equation as a differential equation and solve for the derivative. Then, as proved in my earlier video, if a curve is perpendicular or orthogonal to another, then the slopes of the tangent line must be a negative reciprocal to the tangent line of the other curves. Thus from this fact we can obtain a second differential equation, which luckily is a separable equation, and can be solved resulting in a family of ellipses. This is a very useful example on the steps involved in determining the orthogonal trajectories, which are actually used a lot in physics and engineering applications such as electricity and fluid-dynamics!

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