Families of Curves

4. Straight line with slope and x -intercept equal.

FAMILIES OF CURVES

Obtain the differential equation of the family of plane curves described.

y

1. Straight lines through the origin. y

x x m=a General Equation: y = m(x - a) where m = slope = arbitrary constant General Equation:

Subst. a = m,

y = mx

y = m(x - m)

where m = slope = arbitrary constant Differentiate, y’ = m  m  or m =

m= dy

y=

dx

dy dx

= y’

Substitute,

dx

Substitue m, dy

dy = m dx

y = y’ (x (x - y’) y = xy’ – (y’)2

x

(y’)2 = xy’ – y – y

y dx = x dy y dx – dx – x  x dy = 0

5. Circles with centers at the origin.

2. Straight lines through the fixed point (h,k) ; h and k not to be

y

eliminated.

General Equation : x2 + y2 = r2 2x dx + 2y dy = 0

y

r

x dx + y dy = 0 x

(h,k)

x 6. Circles with center on the x-axis. y

General Equation :

General Equation:

(y - k) = m (x - h)

(x-h)2 + y2 = r2

Where m = slope = arbitrary constant

r

Differentiate,

x

dy

2(x + h) + 2yy’ = 0 x – h + 2yy’ = 0

dx

Differentiate

dy = mdx m=

Differentiate

Substitute, (y - k) =

1 + (yy” + y’y’) = 0

dy

yy” + (y’)2 + 1 = 0

 (x - h)

dx

(y - k) dx = (x - h) dy 6. Parabolas with vertex on the x-axis, with axis parallel to the y-axis,

(y - k) dx - (x - h) dy = 0

and with distance from focus to vertex fixed as ‘a’. 3. Straight lines with slope and y -intercept equal. y

y

x

m=b x General Equation : y = mx + b where m = slope = arbitrary constant

General Equation: (x - h)2 = 4a (y - k)

Substitute b = m,

Subst. k = 0

Y = mx + m

(x - h)2 = 4ay

Differentiate,

h = arbitrary constant

dy = m dx

Differentiate,

m=

dy

2(x - h) = 4ay’

dx

(x - h) = 2ay’

Substitute, y=

dy dx

x+

dy dx

EQ1

(x - h)2 = 4a2 (y’) 2 Subst. in EQ1

y dx = x dy + dy

4a2 (y’) 2 = 4ay

y dx - (x + 1) dy = 0

a(y’) 2 = y