Problem:
Find the orthogonal trajectories to the curves $ y = x^2 + c $.
Solution:
First, we express the parabolas as differential equations
$ \frac{dy}{dx} = 2x $
Next, the slope of the tangent lines for the orthogonal trajectories is determined, so we solve this
$ \frac{dy}{dx} = \frac{-1}{2x} $
This is separable, and therefore the solution is simply
$ y = \frac{-1}{2}sgn(x)\ln|x| + c $
Find the orthogonal trajectories to the curves $ y = x^2 + c $.
Solution:
First, we express the parabolas as differential equations
$ \frac{dy}{dx} = 2x $
Next, the slope of the tangent lines for the orthogonal trajectories is determined, so we solve this
$ \frac{dy}{dx} = \frac{-1}{2x} $
This is separable, and therefore the solution is simply
$ y = \frac{-1}{2}sgn(x)\ln|x| + c $
No comments:
Post a Comment