Name:
Instructor:
Math 10560, Worksheet 17 Direction Fields and Euler’s Method
February 26, 2016
•
•
•
•
Please show all of your work for all questions both MC and PC
work without using a calculator.
Multiple choice questions should take about 4 minutes to complete.
Partial credit questions should take about 8 minutes to complete.
PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
1. (a)
(b)
(c)
(d)
(e)
2. (a)
(b)
(c)
(d)
(e)
........................................................................................................................
3. (a)
(b)
(c)
(d)
(e)
4. (a)
(b)
(c)
(d)
(e)
........................................................................................................................
5. (a)
(b)
(c)
(d)
(e)
6. (a)
(b)
(c)
(d)
(e)
........................................................................................................................
7. (a)
(b)
(c)
(d)
(e)
8. (a)
(b)
(c)
(d)
(e)
Please do NOT write in this box.
Multiple Choice
9.
10.
Total
Name:
Instructor:
Multiple Choice
1.(6 pts)A certain interest rate in the economy, denoted by r, changes with time according
to the differential equation
dr
= 0.1(5 − r).
dt
If this rate is equal to 3 today, use Euler’s method with a stepsize h = 2 to estimate its
value in 4 years from now.
(a)
3.72
(b)
3.4
(c)
1.8
(d)
1.5
(e)
3.5
2.(6 pts)Consider the initial value problem
(
y 0 = sin[π(x + y)]
y(0) = 0.
Use Euler’s method with two steps of step size 0.5 to find an approximate value of y(1).
Note: The formula sheet may help.
(a)
0
(b)
1
(c)
−0.5
2
(d)
0.5
(e)
−1
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Instructor:
3.(6 pts) Use Euler’s method with step size 0.2 to estimate y(0.4) where y(x) is the
solution to the initial value problem
y 0 = 10(x + y)2 ,
y(0) = 0.
Here F (x, y) = 10(x + y)2 , h = 0.2 and the initial point is (0, 0). Therefore,
y1 = y0 + hF (x0 , y0 ) = 0 + 0.2 · F (0, 0) = 0.
Now,
y2 = y1 + hF (x1 , y1 ) = 0 + 0.2 · F (0.2, 0) = 0.2 · 10(0.2)2 = 0.2 · 0.4 = 0.08.
(a)
0.8
(b)
0
(c)
0.08
(d)
0.4
(e)
2.8
4.(6 pts) Use Euler’s method with step size 0.5 to estimate y(1.5) where y(x) is the
solution to the initial value problem
y 0 = y 2 + 2x,
(a)
5
(b)
6
(c)
2
y(0.5) = 1.
(d)
3
8.5
(e)
1
Name:
Instructor:
5.(6 pts) Which of the following gives the direction field for the differential equation
dy
= y2 − 1 ?
dx
Note the letter corresponding to each graph is at the lower left of the graph.
4
Name:
Instructor:
(a)
(b)
(c)
(d)
(e)
6.(6 pts) Which of the following gives the direction field for the differential equation
y 0 = y 2 − x2 .
For points on the line y = x, we must have y 0 = 0. Also for points on the line y = −x,
we must have y 0 = 0. Hence along both diagonals f the plane, we must have y 0 = 0 and
the answer must be (e).
5
Name:
Instructor:
(a)
(b)
(c)
(d)
(e)
6
Name:
Instructor:
7.(6 pts) Use Euler’s method with step size 0.1 to estimate y(1.2) where y(x) is the
solution to the initial value problem
y 0 = xy + 1
y(1) = 0.
x0 = 1, y0 = 0
x1 = x0 + h = 0.1, y1 = y0 + h(x0 y0 + 1) = 0 + (0.1)(1 · 0 + 1) = 0.1
x2 = x1 + h = 0.2, y2 = y1 + h(x1 y1 + 1) = 0.1 + (0.1)((0.1)2 + 1)
= 0.1 + 0.1(0.11 + 1) = 0.1 + 0.1(1.11) = 0.1 + 0.111 = 0.211
(a)
y(1.2) ≈ .112
(b)
y(1.2) ≈ .201
(d)
y(1.2) ≈ .111
(e)
y(1.2) ≈ .211
(c)
y(1.2) ≈ .101
8.(6 pts) Use Euler’s method with step size 0.5 to estimate y(2) where y(x) is the solution
to the initial value problem
y 0 = x(y − x),
y(1) = 2.
Euler’s method gives us the following approximations.
x y(x) y 0 (x) y(x) + 0.5y 0 (x)
1
2
1
2.5
1.5 2.5
1.5
3.25
2 3.25
(a) 2.125
(b)
2
(c)
3
(d)
7
2.75
(e)
3.25
Name:
Instructor:
Partial Credit
You must show your work on the partial credit problems to receive credit!
9. (12 pts.) (a) Which of the pictures below show the direction field for the differential
equation
dy
= (4 − y)(4 + y).
dx
Circle the label at the lower left of your answer to indicate your choice. Justify your
answer with some calculations; enough to distinguish your choice from the other options.
Note that the point (0, 0) is in the center of each picture.
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
(I)
(III)
-6
-6
-4
-2
0
2
4
6
(II)
-6
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-6
-4
-2
0
2
4
6
(IV)
8
-4
-6
-2
-4
0
-2
2
0
4
2
6
4
6
Name:
Instructor:
(b) On the direction field you have selected above , sketch the graph of the solution with
initial condition y(0) = 32 .
(c) For the solution you have sketched in part (b), use the direction field to determine
lim y(x)?
x→∞
9
Name:
Instructor:
dy
= (4 − y)(4 + y) < 0 so the slopes should be negative for all points
dx
dy
= (4 − y)(4 + y) < 0 so all points below y = −4
above y = 4. Similarly when y < −4,
dx
dy
should also be negative. When −4 < y < 4,
= (4 − y)(4 + y) > 0 so all points in
dx
between should have positive slope. This is answer (IV).
(a) When y > 4,
3
(b) The point (0, ) is in the middle portion y should slowly curve up to y = 4.
2
10
Name:
Instructor:
(c) This means that for this initial condition
lim y(x) = 4
x→∞
11
Name:
Instructor:
10. (12 pts.) (a) Sketch the direction field for the differential equation
dy
= 1 − y.
dx
(b) On this sketch, draw the graph of the solution with initial condition y(0) = 32 .
(c) For this solution, what is lim y(x)?
x→∞
Solution. (c) Observe from the graph lim y(x) = 1.
x→∞
4
2
-4
2
-2
-2
-4
12
4
Name:
Instructor:
The following is the list of useful trigonometric formulas:
sin2 x + cos2 x = 1
1 + tan2 x = sec2 x
1
sin2 x = (1 − cos 2x)
2
1
cos2 x = (1 + cos 2x)
2
sin 2x = 2 sin x cos x
1
sin(x − y) + sin(x + y)
2
1
sin x sin y =
cos(x − y) − cos(x + y)
2
1
cos x cos y =
cos(x − y) + cos(x + y)
2
Z
sec θ = ln | sec θ + tan θ| + C
sin x cos y =
13
Name:
Instructor: ANSWERS
Math 10560, Worksheet 17 Direction Fields and Euler’s Method
February 26, 2016
•
•
•
•
Please show all of your work for all questions both MC and PC
work without using a calculator.
Multiple choice questions should take about 4 minutes to complete.
Partial credit questions should take about 8 minutes to complete.
PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
1. (•)
(b)
(c)
(d)
(e)
2. (a)
(b)
(c)
(•)
(e)
........................................................................................................................
3. (a)
(b)
(•)
(d)
(e)
4. (•)
(b)
(c)
(d)
(e)
........................................................................................................................
5. (a)
(b)
(c)
(•)
(e)
6. (a)
(b)
(c)
(d)
(•)
........................................................................................................................
7. (a)
(b)
(c)
(d)
(•)
8. (a)
(b)
(c)
(d)
(•)
Please do NOT write in this box.
Multiple Choice
9.
10.
Total