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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Recall from my earlier video that an orthogonal trajectory of a family of curves is a curve that intersects each curve of the family orthogonally, that is, at right angles.
For instance, each member of the family y = mx of straight lines through the origin is an orthogonal trajectory of the family x2 + y2 = r2 of concentric circles with the origin as the center.
We say that the two families are orthogonal trajectories of each other.
Find the orthogonal trajectories of the family of curves x = ky2, where k is a constant.
The first step is to find a single differential equation that is satisfied by all members of the family:
This differential equation depends on k, but we need an equation that is valid for all values of k simultaneously.
This means that the slope of the tangent line at any point (x, y) on one of the parabolas is y' = y/(2x).
Recall that on an orthogonal trajectory the slope of the tangent line must be the negative reciprocal of this slope on the above family of curves.
Thus the orthogonal trajectories must satisfy the differential equation:
This differential equation is separable and we can solve it as follows:
Thus, the orthogonal trajectories are the family of ellipses given by the above equation and if we graph them out we get:
Note: If C is a negative constant, then it is not a real number because x2 + y2/2 is always positive.
Also, orthogonal trajectories occur in various branches of physics.
For example, in an electrostatic field the lines of force are orthogonal to the lines of constant potential.
Also in streamlines in aerodynamics are orthogonal trajectories of the velocity-equipotential curves.
