Review of Diff Eq and Slope Fields/Euler`s Method 1. Solve the

Review of Diff Eq and Slope Fields/Euler's Method
1. Solve the following differential equations by using the
separation of variables method.
dy
 x2  x
(a)
dx
dy
y

(b)
dx x 2
dy
 y 3 with the initial condition y(0) = 1
(c)
dx
dy
 xy  y
(d)
dx
7. Drawing a slope field
(A) provides a way of visualizing the solution to a
differential equation
(B) can help find horizontal asymptotes to the graph of
the solution of the differential equation
(C) can serve as a check to the solution of a differential
equation
(D) can give evidence as to the symmetry of the graph of
the solution to a differential equation
(E) all of the above
dy
 x2
dx
(A) has line segments symmetric to the y-axis
(B) shows that the solutions to the differential equation
are even functions
(C) shows that the graphs of the solutions are increasing
for increasing x
(D) shows that the graphs of the solutions are decreasing
for increasing x
(E) shows that there are solutions that have a horizontal
asymptote
8. The slope field for the differential equation
2. The differential equation
(A)
dy
 dx
xy  2 y
dy
 xy  2 y
dx
dy
(E)
  x  2  dx
y
(C)
3. A possible solution to
(A) y  5  Ce t
(C) y  5et
dy
 xy  2 y in separable form is
dx
(B) dy   xy  2 y  dx
(D) dy   x  2  y dx
dy
 y  5 is
dt
(B) y  Ce t  5
dy
 y shows that the solutions to the
dx
differential equation
(A) have y-intercept (0, 1)
(B) have a positive y-intercept
(C) have a horizontal asymptote
(D) are even functions
(E) are odd functions
9. The slope field for
(D) y 2  5 y  t  C
(E) y  Ce 5t
4. The solution to the differential equation
P(0) = 5 is
(A) P  5e0.02t
(C) P  5  e 0.02t
1
(E) P 
5  0.02t
dP
 0.02 P with
dt
(B) P  5e 0.002t
(D) P  0.02 P 2  C
5. A radioactive element has a half-life of 1,000 years. (This
means that the amount present initially will be halved
after 1,000 years.)
(a) If 100 grams of the element are present initially, how
many grams remain after 5000 years?
(b) If the rate of change of the number of grams present is
proportional to the amount present, write and solve a
differential equation that represents the amount y of the
element remaining after t years.
6. In a slope field, the line segments are
(A) part of the graph of the solution to the differential
equation
(B) parts of the lines tangent to the graph of the solution
to the differential equation
(C) asymptotes to the graph of the solution of the
differential equation
(D) lines of the symmetry of the graph of the solution to
the differential equation
(E) none of the above
10.
dy
 2 xy
dx
(a) Sketch a slope field for points in the plane with integer
values of x and y between -2 and 2 inclusive.
(b) Use the slope field to determine whether the solutions
to the differential equation are even, odd, or neither.
(c) Use the slope field to determine whether the solutions
to the differential equation have a horizontal asymptote.
(d) Sketch a solution on the slope field that passes through
the point (0, 1).
11. The solution to
dy
 y cos x is
dx
(A) y  Ce sin x
(B) y  sin x  C
(C) y   sin x  C
(D) y  e sin x  C
(E) y  Ce  sin x
dy
  x with initial condition y(0) = 1
dx
(A) is always concave up
(B) is always concave down
(C) is undefined at x = 0
(D) is always increasing
(E) is always decreasing
12. The solution to
13. Which is the slope field for the differential equation
dy
 2y  4 ?
dx
(A)
(B)
16. The differential equation
dy
 2 y  50 written in
dx
separable form is
1 dy
(A)
 50
2 y dx
(B)
dy
 2 dx
y  50
(C) dy   2 y  50  dx
(D)
dy
 2 dx
y  25
(E)
dy
 50 dx
2y
17. Which of the following could be a solution of the
differential equation with the given slope field?
(C)
(D)
(A) y  x  1
(B) y  x 2  2
(C) y  x 3  2
14. Which graph shows a slope field with a solution to the
dy
 y2 ?
differential equation
dx
(A)
(B)
(C)
(D)
(D) y  ln  x  1
(E) y  2e x
18. Which of the following could be a solution of the
dy
 1  x 2 1  y 2  ?
differential equation
dx
(A) y  x


x3
(B) y  sin  x    
3


(C) y  3 x 
15. Of the four curves A, B, C, and D on the slope field
shown, which graphs the function that is the solution to
dy
x
the equation
  with initial condition y(0) = 4?
dx
y
(A) A
(D) D
(B) B
(E) none
(C) C
x3
6
3

x3 
(D) y  arctan  x  
3 



x3
(E) y  tan  x   6 
3


19. The solution to the differential equation
initial condition y(0) = 9 is
9
(A) y 
2
3
 x  1
 x3

(C) y    3 
 6

(E) y  e


9 x3 1
dy
 x 2 y with
dx
9
(B) y  3
x 1
dy
 2 y , with the initial
dt
condition y(0) = 10.
dy
 x  2 with the initial
dx
point (0, 1), use Euler’s method with a step size of 0.1 to
approximate the value of f  0.2  .
25. Given the differential equation
2
3
(D) y  e x 1
2
dy
 3x with the initial
dx
condition f(1) = 1, use Euler’s method with a step size of
0.1 to approximate the value of f 1.2  .
26. Given the differential equation
20. A student in an SAT verbal class learns 4 vocabulary
words the first day. The student’s learning curve is given
dw
 0.2  w  8 , where w
by the differential equation
dt
is the number of words learned. If this learning curve is
correct, what is the maximum number of words the
student can learn per day?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
21. If
24. Solve the differential equation
dy
ty

and y  4   e 2 , then y =
dt
ln y
1
(A) y  e 2
 20t 
2
20t 2
(B) y  e
1
(C) y  e 2
20 t 2
27.
dy
x

and y(0) = 0. Using Euler’s method with step
dx 1  x 2
x  0.2 , y(0.2) is approximately
(A) 0
(B) 0.0196
(C) 0.192
(D) 0.2
(E) 0.888
28.
dy
 x  y and y(0) = 1. Using Euler’s method with step
dx
x  0.5 , approximate the value of y(0.5).
(A) 0
(B) 1
(C) 1.5
(D) 1.797
(E) 2
(D) y  ln 20  t 2
(E) y  eln
22.
dy
 x  2 xy and y  0   1 . Find y as a function of x.
dx
2
(A) y  3e x  2
(B) y  3 x 2  1
3 2 1
(C) y  e x 
2
2
(D) y  3 ln  x 2  1  1

x2
initial condition y(1) = 4.
 y  1 and y(0) = 3. What is the approximate
 dy
dx
30. x 2  1
3
1
(E) y  e 2 
2
2
23. Solve the differential equation y
dy
 y  1 and y(1) = 2. Use Euler’s method with step
dx
x  0.1 to approximate y(1.2).
(A) 2
(B) 2.091
(C) 2.1
(D) 2.181
(E) 2.191
29. x 2
20t 2
dy
 2t  7 , with the
dt
value of y(0.1) using Euler’s method with step x  0.1 ?
(A) -0.891
(B) 3
(C) 3.2
(D) 3.21
(E) 3.418
Answers:
1
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
x3 x 2
(a) y    C
3
2
(b) y  Ce
(c) y 

1
x
1
1  2x
(d) y  Ce
2. E
3. A
4. (A)
5. (a) 3.125 g
x2
x
2
(b) y  100e6.9310
6. B
7. E
8. C
9. C
10. (a)
23. y  2t 2  14t
4
24. y  10e 2 t
t
25. f  0.2   1.41
26. f 1.2   1.63
27. A
28. C
29. E
30. C
y
3
2
1
x
-2
-1
1
2
-1
-2
-3
(b) even
(c) no horizontal asymptote
(d)
y
3
2
1
x
-2
A
B
A
B
B
D
B
E
C
E
B
C
-1
1
-1
-2
-3
2