SLOPE FIELDS ASSIGNMENT #15 NAME_________________

31

1.

The calculator drawn slope field for the

differential equation dy xydx

= is shown in the

figure below. The solution curve passing through the point (0, 1) is also shown.

2. The calculator drawn slope field for the differential

equation dy x ydx

= + is shown in the figure below.

(a) Sketch the solution curve through the point (0,

2).

(a) Sketch the solution curve through the point (0, 2).

(b) Sketch the solution curve through the point (0, -

1)

(b) Sketch the solution curve through the point (0, -2)

Draw a slope field for each of the following differential equations. Show a segment at each indicated point.

3. 1dy

xdx

= + 4. 2dyy

dx= **Note the

scales!!!

5. dy ydx x

= −

(a) Sketch a solution curve

which passes through the

point (1, 0)

(a) Sketch a solution curve which

passes through the point (0, -

1).

(a) Sketch a solution curve which

passes through the point (2, -1).

For problems 6-8, find the equations of the solution curves you sketched in problems 3-5. Each equation should be

expressed in the form of ( )y f x= . Use your graphing calculator to graph each of your equations for problems 6-8

to see if those graphs match your solution curves drawn in problems 3-5.

−4 −2 2 4

−4

−2

2

4

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

x

y

−4 −2 2 4

−4

−2

2

4

−3 −2 −1 1 2 3

−1.0

−0.5

0.5

1.0

•

SLOPE FIELDS ASSIGNMENT #15 NAME_________________

32

6. 1dy

xdx

= + 7. 2dyy

dx= 8. dy y

dx x= −

9. At right is a slope field for the differential equation

xdye

dx−= .

(a) Sketch the solution curve passing through the point (0, 0)

(b) Find a particular solution in the form of ( )y f x= to the

differential equation xdye

dx−= .

(c) Without the use of a calculator (instead use transformations

to the graph of xy e= ), determine whether your equation

from part b represents the function which you graphed in

part a.

10. The slope field for a differential equation is shown at the

right. Which statement is true for solutions of the

differential equation?

I. For 0x < all solutions are decreasing

II. All solutions level off near the x-axis.

III. For 0y > all solutions are increasing

(a) I only (b) II only (c) III only (d) II and III only (e) I, II and III

SLOPE FIELDS ASSIGNMENT #15 NAME_________________

33

11. The slope field for the differential equation

2 2

4 2dy x y ydx x y

+=

+ will have horizontal segments when

(a) 2 ,y x= only (b) 2 ,y x= − only (c) 2 ,y x= − only (d) 0y = , only (e) 0y = or 2y x= −

12.

Which one of the following could be the graph of the solution of

the differential equation whose slope field is shown at right.

(E) (A) (B)

(C) (D)

13.

Which statement is true about the solutions y(x), of a differential

equation whose slope field is shown at the right.

I. If y (0) > 0 then lim ( ) 0.xy x

→∞=

II. If 2 (0) 0y− < < then lim ( ) 2.xy x

→∞≈ −

III. If y (0) < -2 then lim ( ) 2.xy x

→∞≈ −

(A) I ONLY (B) II ONLY (C) III ONLY (D) II and III ONLY (E) I, II, and III

SLOPE FIELDS ASSIGNMENT #15 NAME_________________

34

14. Shown at the right is the slope field for which of the

following differential equations?

(A) 1dyx

dx= + (B) 2dy

xdx

= (C) dy x ydx

= +

15. Consider the differential equation given by 2

dy xydx

= .

(a) On the axes provided below, sketch a slope field for the given differential equation at the nine points

indicated.

−1 1

1

2

3

x

y

(b) Find the particular solution ( )y f x= to the given differential equation with the initial condition

(0) 3f = .

SLOPE FIELDS ASSIGNMENT #15 NAME_________________

35

16.

Consider the differential equation 2 4dyy x

dx= − .

−1 1

−2

−1

1

2

x

y

(a) The slope field for the given differential equation is

provided. Sketch the solution curve that passes

through the point (0, 1) and sketch the solution curve

that passes through the point (0, -1).

(b) Find the value of b for which 2y x b= + is a solution to the given differential equation. Justify your

answer.

(c) Let g be the function that satisfies the given differential equation with the initial condition g (0) =0.

Does the graph of g have a local extremum at the point (0, 0) ? If so, is the point a local maximum

or a local minimum? Justify your answer.

17. Consider the differential equation given by ( )21dyx y

dx= − .

(a) On the axes provided, sketch a slope field for the given differential equation at the eleven points

indicated.

−2 −1 1 2

−1

1

x

y

(b) Use the slope field for the given differential equation to explain why a solution could not have the

graph shown below.

−2 −1 1 2

−1

1

x

y

SLOPE FIELDS ASSIGNMENT #15 NAME_________________

37

Match each slope field with the equation that the slope field could represent.

(A)

(B)

(C)

(D)

(E) (F)

(G)

(H) 18. 1y =

20. 2

1y

x=

22. 2y x=

24. cosy x=

19. y x=

21. 316

y x=

23. siny x=

25. lny x=

Match the slope fields with their differential equations.

(A)

(B) (C)

(D)

(E) 26. cosdy

xdx

=

28. dy xdx y

= −

30. dy ydx

=

27. dy x ydx

= −

29. 1 12

dyx

dx= +