Difficult

1. Straight lines through the origin.

Solution:

y=mx+c , c = 0

m=yx

y dx-x dyy2=0 ⟹ 1 dxy - x dyy2 =0

1 dxy = x dyy2 ⟹ 1 dxx = 1 dyy ⟹ yx = dydx

Answer: y dx-x dy=0

2. Straight lines with slope and y-intercept equal.

Solution:

y = mx + b, m = b = c

y = cx + c

c= yx+1 ⟹ x+1 dy-y dxx+1=0 ⟹ y dx=x+1 dy

Answer: y dx-x+1 dy=0

3. Straight lines with algebraic sum of the intercepts fixed as k.

Solution:

For x-intercept:

y = m(x – a)

y’ = m ⟹ y = y’ (x – a)

y = xy’ – ay’ ⟹ a = (xy’ – y)/y’

For y- intercept:

y = mx + b

y’ = m ⟹ Y = y’x + b

b = y –xy’

But, k = a + b

K = (xy’ – y)/y + (y – xy’)

Multiply by y’,

ky’ = xy’ –y + y’ y-xy’ ⟹ ky’= (1 – y’)(xy’ – y)

Answer: xy’ – yy’ - 1+ky’ = 0

4. Circles with center at the origin.

Solution:

x2+y2= r2

2x dx+2y dy=0 ⟹ dydx=-xy

Answer: x dx+y dy=0

5. Circles with fixed radius r and tangent to the x-axis.

Solution:

(x-a)2+(y+r)2=r2

2x-a+2y+r y'=0 ⟹ 2x-a=-2y+ry'

x-a=-y+r y'

-y+ry'2+ y+r2=r2

Answer: y±r2(y')2+y2±2ry=0

6. Circles with center on the line y= -x, and passing through the origin.

7. All circles. Use the curvature.

8. Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with distance from focus to vertex fixed as a.

Solution:

(y-k)2=4ax

2y-bdydx = 4a ⟹ y-bdy dx= 2a 

y-b2 dydx2 = 4a2

4ax dydx2=4a2 

Answer: x dydx2=a or x (y')2=a

9. Parabolas with axis parallel to the x-axis and with distance from vertex to focus fixed as a.

10. Use the fact that

d2xdy2= ddy dxdy= dxdy ddxdxdy= dxdy ddxdydx-1= -y''y'3

to prove that the answers to Exercise 17 and 18 are equivalent.

11. Parabolas with axis parallel to the x-axis.

12. The confocal central conics

x2a2+ λ+ y2b2+ λ=1

with a and b held fixed.

13. The cubics of Exercise 24 with c held fixed and a to be eliminated.

Solution:

cy2=x2(x-a)

2cyy'=3x2-2ax ⟹ y'=x(3x-2a)2cy

14. The quartics...