Working with orthogonal trajectories
(a)Find the orthogonal trajectories of the family of curves defined by
2cy + x2 = c2, c>0
State the differential equation of the orthogonal family, and show your steps in obtaining a solution.
(b) On the same set of "square" axes, plot at least five members of each of the given family and your family of orthogonal solutions.
(c) Identify one point of intersection, and approximate the coordinates of the point. Determine the slope of each curve at the point of intersection, and verify that the tangents are indeed perpendicular.
NOTE: So far, I've done implicit differentiation, and gotten 2cdy/dx + 2x = 0. Then I solved for c (from the original equation) by using the quadratic formula. But when I plug c back in and try to solve the differential equation (I've been getting a homogeneous differential equation and substitute x=vy), it doesn't seem to turn out right. Either the D.E. is too hard to integrate; or I can't get the tangents to be perpendicular in the end.
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2cy + x2 = c2, c>0
Solution
The level curves constitute a family of curves, one for each value of the constant, and this family can be described by a single differential equation. We find the DE by eliminating the constant. This is ...
Solution Summary
The orthogonal trajectories for differential equations are provided. Clear explanations and all mathematical steps are provided.