03.1 orthogonal treajectories (1)
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1) An orthogonal trajectory of a family of curves is a curve that intersects each curve of the family orthogonally, or at right angles. 2) To find the orthogonal trajectories of a given family of curves, you first find the differential equation for the family of curves. Then you derive the differential equation for the orthogonal trajectories by taking the inverse of the original equation. 3) Solving the new differential equation provides the equation for the family of orthogonal trajectories.
03.1 orthogonal treajectories (1)
- 3. • An orthogonal trajectory of a family of curves is a curve that intersects each curve of the family orthogonally—that is, at right angles.
- 5. METHODMETHOD Given family of curve ( , , )F x y c o= STEP 1 Find the differential equation for the given family of curves, by differentiating Eq.1. ( , ) dy f x y dx = STEP 2 Find the differential equation of the Orthogonal Trajectory 1 ( , ) dy dx f x y = − STEP 3 Eq.3. Eq.2. Eq.1. Solution of Eq.3. will be the equation of the family of orthogonal trajectories ( , , )G x y c o= OrthogonalTrajectoriesChapter3
- 6. Orthogonal Curves (1) = =By differentiation we get: . Hence the family of parabola in question satisfies the differential equation 2 . 2 dy x dx dy x dx 2 Consider the family of parabola . Find the family of curves which intersect the above family of parabola perpendicularly. y x C= + Example Solution Two curve intersect perpendicularly if the product of the slopes of the tangents at the intersection point is -1. The differential equation for the orthogonal family of curves. = − 1 2 dy dx x OrthogonalTrajectoriesChapter3
- 7. Orthogonal Curves (2) = − ⇒ = − ⇒∫ ∫ 1 1 1 2 2 dy dy dx dx x x It remains to solve 2 Consider the family of parabola . Find the family of curves which intersect the above family of parabola perpendicularly. y x C= + Example Solution (cont’d) The figure on the right shows these two orthogonal families of curves. = − 1 2 dy dx x = − + 1 ln 2 y x C OrthogonalTrajectoriesChapter3
- 8. Describe the orthogonal trajectories for the family of curves given by xy = C Solution y2 – x2 = K y2 – x2 = K
- 10. Find the orthogonal trajectories of the family of curves x = ky2 , where k is an arbitrary constant. 1 1 2 or = 2 dy dy ky dx dx ky = 2 1 1 2 2 dy xdx ky y y = = Solution
- 11. 2 dy y dx x = 2dy x dx y = − 2 2 2 2 2 2 2 y dy xdx y x C y x C = − = − + + = ∫ ∫