Top Banner
GEC BHAVNAGAR
13

03.1 orthogonal treajectories (1)

Aug 19, 2014

Download

Engineering

jay shah

maths
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: 03.1 orthogonal treajectories (1)

GEC BHAVNAGAR

Page 2: 03.1 orthogonal treajectories (1)

O

rthog

onal

Tra

ject

orie

s

Page 3: 03.1 orthogonal treajectories (1)

• An orthogonal trajectory of a family of curves is a curve that intersects each curve of the family orthogonally—that is, at right angles.

Page 4: 03.1 orthogonal treajectories (1)

Ort

hogo

nal T

raje

ctor

ies

Page 5: 03.1 orthogonal treajectories (1)

METHODMETHODGiven 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 ydx

STEP 2 Find the differential equation of the Orthogonal Trajectory

1( , )

dydx 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

Orth

ogon

al T

raje

ctor

ies

Chap

ter 3

Page 6: 03.1 orthogonal treajectories (1)

Orthogonal Curves (1)

By differentiation we get: . Hence the family of parabola

in question satisfies the differential equation 2 .

2

dy xdx

dy xdx

2Consider the family of parabola . Find the family of curveswhich 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

2dydx xOr

thog

onal

Tra

ject

orie

s Ch

apte

r 3

Page 7: 03.1 orthogonal treajectories (1)

Orthogonal Curves (2)

1 1 1

2 2dy dy dxdx x x

It remains to solve

2Consider the family of parabola . Find the family of curveswhich 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

2dydx x

1 ln2

y x C

Orth

ogon

al T

raje

ctor

ies

Chap

ter 3

Page 8: 03.1 orthogonal treajectories (1)

Describe the orthogonal trajectories for the family of curves given by

xy = C

Solution

y2 – x2 = K

y2 – x2 = K

Page 9: 03.1 orthogonal treajectories (1)
Page 10: 03.1 orthogonal treajectories (1)

Find the orthogonal trajectories of the family of curves x = ky2, where k is an arbitrary constant.

11 2 or = 2

dy dykydx dx ky

2

1 12 2

dyxdx ky yy

Solution

Page 11: 03.1 orthogonal treajectories (1)

2dy ydx x

2dy xdx y

22

22

2

2

2

y dy x dx

y x C

yx C

Page 12: 03.1 orthogonal treajectories (1)
Page 13: 03.1 orthogonal treajectories (1)