Differential Equation

Chapter 1 (Definition) 1.

9.

2.

10.

3. (

11. (

) ( )

4.

̇

( )

5.

)

(

12. (

)

( )

13. (

)

(

)

)

6. 14. 7.

15. ( )

8.

16.

17.Verify that

is a solution of the equation in Exercise 1.

18. Verify that

is a solution of the equation in Exercise 5.

19. Verify that

is a solution of the equation in Exercise 5.

20. Bessel’s function of index zero is defined by the power series. Verify that ( ) is a solution of the differential equation. ( )

∑

21. Verify that for

(

)

( )

( ) √

is a solution of the equation in Exercise 6.

Chapter 1 (Elimination of Arbitrary Constants) 1.

12.

2.

13.

3.

14.

4.

15.

5.

(

);

16.

a

parameter, not to be eliminated. 6.

;

18.

a

19.

parameter.

20.

7. 8.

17.

;

a parameter,

to

21. ;

be eliminated. parameters.

9. 10. 11.

22. 23. 24. 25.

and

are

Chapter 1 (Families of Curves) 1. Straight lines through the origin. );

2. Straight lines through the fixed point(

and

not to be eliminated.

3. Straight lines with slope and y-intercept equal. 4. Straight lines with slope and x-intercepts equal. 5. Straight lines with algebraic sum of the intercepts fixed as . 6. Straight lines at a fixed distance

from the origin.

7. Circles with center at the origin. 8. Circles with center on the x-axis. 9. Circles with fixed radius and tangent to the x-axis. 10. Circles tangent to the x-axis. 11. Circles with center on the line

, and passing through the origin.

12. Circles of radius unity. Use the fact that the radius of curvature is 1. 13. All circles. Use the curvature. 14. Parabolas with vertex on the x-axis, with axis parallel to the y-axis, and with distance from focus to vertex fixed as . 15. Parabolas with vertex on the y-axis, with axis parallel to the x-axis, and with distance from focus to vertex fixed as . 16. Parabolas with axis parallel on the y-axis and with distance from the vertex to fixed as . 17. Parabolas with axis parallel on the x-axis and with distance from the vertex to fixed as . 18. Work Exercise 17, using differentiation with respect to .

19. Use the fact that (

)

(

)

(

)

( )

to prove that the answers to Exercise 17 and 18 are equivalent. 20. Parabolas with vertex and fous on the x-axis. 21. Parabolas with axis parallel to the x-axis. 22. Central conics with center at the origin and vertices on the coordinate axes. 23. The confocal central conics (

24. The cubics

with and ) with

held fixed.

25. The cubics of Exercise 24 with held fixed and (

26. The quartics

) with

(

29. The cissoids

to be eliminated.

)

.

.

30. The trisectrices of Maclaurin

(

)

(

31. Circles through the intersections of the circle Use the “

to be eliminated.

held fixed.

27. The quartics of Exercise 26 with held fixed and 28. The strophoids

held fixed.

” form; that is, the equation ) and (

32. Circles through the fixed points (

). and the line (

)

34. The cardioids

(

). (

).

35. The cissoids 36. The strophoids

. (See Exercise 29.) (

37. The trisectrices of Maclaurin

). (

.

). Use the method of Exercise

31. 33. The circles

.

). (See Exercise 30.)