How can orthogonal trajectories be found for a specific family of curves?

In summary, the conversation discusses finding the set of orthogonal trajectories for a given family of curves represented by the equation (x-c)^2 + y^2 = c^2. The process involves manipulating the equation and eventually solving a separable differential equation for the variable v as a function of x.
  • #1
intenzxboi
98
0
The set of orthogonal trajectories for the family indicated by
( x-c)^2 + y^2 = c^2

My work:

y' = -(x-c)/y Since c= ( x^2 + y^2 ) / 2x

plugging back in and doing -1/y' i got

y' = 2xy / ( x^2 - y^2)

Then I am supposed to move the x and y to a side and integrate but i don't see how it is possible.
 
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  • #2
intenzxboi said:
The set of orthogonal trajectories for the family indicated by
( x-c)^2 + y^2 = c^2

My work:

y' = -(x-c)/y Since c= ( x^2 + y^2 ) / 2x

plugging back in and doing -1/y' i got

y' = 2xy / ( x^2 - y^2)

Then I am supposed to move the x and y to a side and integrate but i don't see how it is possible.
You can't. This is NOT a "separable" differential equation. It is, however, a "homogeneous" equation because replacing x by a x and y by ay on the right gives 2(ax)(ay)((ax)^2- (ay)^2)= 2a^2xy/(a^2(x^- y^2)= 2xy/(x^2- y^2) again. That essentially means that the right hand side can be written as a function of y/x. Let v= y/x so that y= xv, y'= xv'+ v and replace y in the right side by xv to get a separable equation for v as a function of x.
 

What are orthogonal trajectories?

Orthogonal trajectories are a set of curves that intersect another set of curves at right angles. They are also known as "conjugate" or "perpendicular" curves.

How do you find orthogonal trajectories?

To find orthogonal trajectories, you need to take the derivative of the given curve and then find the negative reciprocal of the derivative. This will give you the slope of the orthogonal trajectory.

What are some applications of orthogonal trajectories?

Orthogonal trajectories have many applications in fields such as physics, engineering, and geometry. They are used to study electric fields, heat flow, and fluid dynamics. They are also used in computer graphics to create three-dimensional images.

Can orthogonal trajectories be found for any set of curves?

No, orthogonal trajectories can only be found for certain types of curves, such as circles, parabolas, and hyperbolas. Curves that do not have a derivative, such as straight lines, do not have orthogonal trajectories.

What is the significance of orthogonal trajectories?

Orthogonal trajectories have a special property where they intersect at right angles. This makes them useful for solving problems involving perpendicular lines or angles. They also provide a way to visualize and understand complex systems and relationships between curves.

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