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Exercises In each exercise, obtain the differential equation of the family of plane curves described and sketch several representative members of the family. 1. Straight lines through the origin. ANS. ydx - xdy = 0. 2. Straight lines through the fixed point (h, k); h and k not to be eliminated. ANS. (y - k) dx - (x - h) dy = 0. 3. Straight lines with slope and y-intercept equal. ANS. ydx - (x + 1) dy = 0. 4. Straight lines with slope and x-intercept equal. ANS. () = xy' - y. 5. Straight lines with algebraic sum of the intercepts fixed as k. ANS. (xy' - y) (y' - 1) + ky' = 0. "\\ 6. Straight lines at a fixed distance p from the origin. ANS. (xy' - y)2 = p2[1 + (v') 2]. 7. Circles with center at the origin. ANS. xdx + y dy = 0. 8. Circles with center on the x-axis. ANS. yy" + ())2 + 1 =0. 9. Circles with fixed radius r and tangent to the x-axis. ANS. (y + r) 2 (y')2 + y2 + 2ry = 0. 10. Circles tangent to the x-axis. ANS. [1 + (y')2]3 = [yy" + 1+(y')2]2. 11. Circles with center on the line y = - x, and passing through the origin. ANS. (x2 - 2xy - y2 ) dx + (x2 + 2xy - 2) dy = 0. 12. Circles of radius unity. Use the fact that the radius of curvature is 1. ANS. (v")2 = [1 + (v')2]3. 13. All circles. Use the curvature. ANS. y"[1 + ()') ] = 3y'(v')2. 14. Parabolas with vertex on the x-axis, with axis parallel to the y-axis, and with distance from focus to vertex fixed as a. ANS. a(y')2 = y. 15. Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with distance from focus to vertex fixed as a. ANS. x()') = a. 16. Parabolas with axis parallel to the y-axis and with distance from vertex to focus fixed as a. ANS. 2ay" = 1 17. Parabolas with axis parallel to the x-axis and with distance from vertex to focus fixed as a. ANS. 2ay" + (y')3 = 0. 18. Work exercise 17, using differentiation with respect to y. ANS. d' x 2adv2 : 1. 19. Use the fact that d' x dyz d dx dx d dx dx d (dy -1 dy dy dy dx dy dy dx dx to prove that the answers to exercises 17 and 18 are equivalent