Download mat 275 test 1 practice and more Summaries Differential Equations in PDF only on Docsity!
MAT 275 TEST 1 PRACTICE
1. Determine if each of the following equations is separable (Yes or No), and /or linear (Yes or No).
Record your answer in the following table. Do not attempt to solve the equations.
Equation Separable Linear
′
′
′
= cos(𝑡𝑦)
′
3
2. a) Write down a first order linear ODE whose solutions all approach 𝑦 = 1.
b) Write down a first order linear ODE such that solutions other than 𝑦 = − 3
all diverge from 𝑦 = − 3.
3. Consider the ODE 𝑦
′
2
a) Determine all the equilibrium (constant) solutions and classify them as stable, unstable or semi-stable.
b) If 𝑦
= − 1 , what will be the behavior of the solution as 𝑡 → ∞?
c) If 𝑦( 0 ) = 0 , what will be the behavior of the solution as 𝑡 → ∞?
4. Which of the following differential equations best
represents the slope field at the right?
a) 𝑦
′
b) 𝑦
′
c) 𝑦
′
d) 𝑦
′
e) 𝑦
′
𝑥
𝑦
5. Consider the direction field below.
a) i. Sketch the graph of the solution which has initial value y (0) = 1.5.
ii. Sketch the solution which has initial value y (0) = 0.5.
b) Use your sketch to estimate the value of y (1) for the solutions in (a) i. and (a) ii.
c) Some solutions of this differential equation grow when t is large and some do not. On the graph,
sketch the curve representing the boundary between these two behaviors. Continue it in both directions
until it leaves the direction field box. Label this curve “isocline”.
d) What happens to the two solutions in part (a) as 𝑡 → ∞?
e) Consider the solution with initial condition y (0) = 1. Estimate the value of y (50). Explain your reasoning.
f) The direction field above comes from one of the following differential equations.
Tell which one it is and why.
I. 𝑦
′
II. 𝑦
′
−𝑡
III. 𝑦
′
d)
𝑥
2
𝑦
2
− 3
𝑑𝑦
𝑑𝑥
1
2 𝑦
e)
𝑑𝑦
𝑑𝑥
4
13. A tank initially contains 60 gal of pure water. Brine containing 1 lb of salt per gallon enters the tank at
2 gal/min, and the (perfectly mixed) solution leaves the tank at 3 gal/min; thus the tank is empty after
exactly 1 hour. Let y ( t ) be the amount of salt in the tank after t minutes.
(a) Write an Initial Value Problem for the amount of salt in the tank at any time t (< 60).
(b) Solve the IVP in part (a) to find the amount of salt in the tank at any time t (< 60).
(c) Determine the amount of salt when the tank is half empty.
14. A completely filled 20 gallon tank originally contains 10 pounds of salt dissolved in water.
Pure water enters the tank at the rate of 5 gallons/minute, and the well-stirred mixture leaves the tank
at the same rate. Find the amount of salt in the tank at any time t.
15. A field mouse population satisfies the Initial Value Problem:
𝑑𝑝
𝑑𝑡
(a) Find the time at which the population becomes extinct.
(b) Find the time at which the population becomes extinct if the initial condition is 𝑝
16. Newton's law of cooling is 𝑢
′
= −𝑘(𝑢 − 𝑇) where u ( t ) is the temperature of an object, t is in hours, T is a
constant ambient temperature, and k is a positive constant. Suppose a building loses heat in accordance with
Newton's law of cooling. Suppose that the rate constant k has the value 0 .13ℎ𝑟
− 1
. Assume that the interior
temperature of the building is 76°F when the heating system fails and the external temperature is T=10°F.
(a) How long will it take for the interior temperature to fall to 32°F?
(b) What happens to the temperature u ( t ) as 𝑡 → ∞?
17. The population of a city increases continuously at a rate proportional, at any time, to the population at
that time. The population doubles in 50yr. Find the ratio of the population P to the initial population P
0
after 75 years.
18. Suppose P(t) denotes the size of an animal population at time t and its growth is described by the differential
equation
𝑑𝑃
𝑑𝑡
= 0. 00 𝑃( 1000 − 𝑃). Determine the value of P at which the population is growing fastest.
19. Solve the following Initial Value Problems using the method of integrating factor.
a) 𝑦
′
2
𝑡
4
b) 𝑦
′
5 𝑡
2 𝑡
c) 𝑡𝑦
′
sin(𝑡)
𝑡
d)
′
e)
𝑑𝑦
𝑑𝑡
20. Determine (without solving the problem) the maximal interval in which the solution of the given initial value
problem is guaranteed to exist:
′
- tan(𝑡) 𝑦 = sin(𝑡) , 𝑦(𝜋) = 6.
21. a) Verify that both 𝑦
1
= 2 𝑡 − 1 and 𝑦
2
2
are solutions to 𝑦
′
2
In which intervals in t are the solutions valid?
b) Does the existence of two solutions of the given problem contradict any known theorem about
existence and uniqueness of solutions to Differential Equations?
22. Consider the following differential equations. Determine if the Existence and Uniqueness Theorem does or
does not guarantee existence and uniqueness of a solution of each of the following initial value problems.
I.
𝑑𝑦
𝑑𝑥
II.
𝑑𝑦
𝑑𝑥
III. 𝑦
𝑑𝑦
𝑑𝑥
IV. 𝑦
𝑑𝑦
𝑑𝑥
23. Use Euler's method and two steps with ∆𝑥 = 0. 1 for the differential equation y ' = y , with initial value
𝑦( 0 ) = 1 , to find the approximate value of 𝑦( 0. 2 ).
24. Use Euler's method and three steps with ∆𝑡 = 0. 2 for the differential equation 𝑦
′
2
− 2 , with
initial value 𝑦( 1 ) = 1 , to find the approximate value of 𝑦( 1. 6 ).
25. Which statement about Euler's method is false?
I. If you halve the step size, you approximately halve the error.
II. Euler's method never gives exact solutions.
III. Euler's method assumes that the slope of a solution curve is the same at all points in a short interval.
IV. Often, when applying Euler's method, the more steps you take the smaller the error.
V. Euler's method is used to string together a set of linearizations that approximate the curve.
26. Find a real valued solution to the following initial value problems. Sketch a graph of the solution.
a) 𝑦
′′
′
+ 2 𝑦 = 0 , with 𝑦
′
b) 2 𝑦
′′
′
− 2 𝑦 = 0 , with 𝑦( 0 ) = 1 , 𝑦
′
c) 𝑦
′′
− 5 𝑦 = 0 , with 𝑦( 0 ) = 1 , 𝑦
′
d) 3 𝑦
′′
− 4 𝑦′ = 0 , with 𝑦( 0 ) = 2 , 𝑦
′
Answers:
st
DE: Yes, No ; 2
nd
DE: Yes, Yes; 3
rd
DE: No, No; 4
th
DE: No, Yes
2. Possible answers: (a) y ' = 1− y (b) y ' = y + 3
3. (a) y = – 2 stable, y = 0 semi-stable, y = 1 unstable
(b) lim
𝑡→∞
= − 2 (c) lim
𝑡→∞
4. (a)
5. (b) i. 𝑦( 1 ) ≈ 3. 3 ii. 𝑦( 1 ) ≈ 0. 6
(d) The first solution approaches ∞ ;
The second solution approaches – ∞
(e) 51 (f) III.
6. (a) y = 0 stable; y = 3 semi-stable; y = 6 unstable; y = 8 stable (b) [0,3)
7. (a) second order (b) linear (c) A = 2
8. (a) first (b) non linear (c) 𝐴 = ±√ 2
9. r =3, r = – 2 10. = −
2 11. d)
a) 𝑦 = − 1 − √ 2 𝑥
2
− 1 Interval: (
1
√
2
b) 𝑦 =
− 1
ln
( 𝑥
) −
3
𝑥
8
3
− 3 𝑥
3 𝑥ln
( 𝑥
) − 9 + 8 𝑥
Interval: ( 0 , 1. 089 )
c) 𝑦 = −√ 2 ln
2
Interval:
d) 𝑦 =
1 −
1
𝑥
Interval:
e) 𝑦 = 4 𝑒
4 𝑥
5
Interval:
13. (a)
𝑑𝑦
𝑑𝑡
3 𝑦
60 −𝑡
(b) 𝑦(𝑡) = 60 − 𝑡 −
( 60 −𝑡)
3
3600
(c) 𝑦
= 22. 5 lbs
−𝒕/𝟒
15. (a) 𝑡 = 4 ln
≈ 2. 77 (b) The population will never become extinct.
16. (a) 8.45 hrs (b) lim
𝑡→∞
𝑢(𝑡) = 10 ( u ( t ) will approach the external temperature).
(a) 𝑦(𝑡) = 𝑡
2
3
− 5 ) (b) 𝑦(𝑡) = (
2
3
3 𝑡
11
3
2 𝑡
(c) 𝑦
cos
( 𝑡
) +𝜋
2
𝑡
2
(d) 𝑦
2
(e) 𝑦(𝑡) = 25 − 19 𝑒
−𝑡
2
/ 10
21. (a) 𝑦
1
(𝑡) is a solution for 𝑡 ≥ 1 ; 𝑦
2
(𝑡) is a solution for all 𝑡;
(b) 𝑓
𝑦
is not continuous at (1, 1).
22. Only II and III are guaranteed to have a unique solution.
25. II
26. (a) 𝑦
− 2 𝑡
−𝑡
(b) 𝑦
𝑡/ 2
− 2 𝑡
(c) 𝑦(𝑡) =
5 −√ 5
10
√ 5 𝑡
5 +√ 5
10
−√ 5 𝑡
(d) 𝑦(𝑡) = 8 + 6 𝑒
4 𝑡/ 3
27. (a) 1 < 𝑥 < 3 (b) −
𝜋
2
< 𝑡 < 1 (c) 0 < 𝑡 < 4
28. II
29. A, B, E, G, I, J
30. C: 𝑡𝑒
𝑡
is not a solution; D: the two functions are not linearly independent
31. By the principle of superposition, B, C, D
32. (i) 𝑡 > 0
(iii) 𝑊
1
2
1
𝑡
3
. Since the Wronskian is nonzero on I, 𝑦
1
(𝑡) and 𝑦
2
(𝑡) form a fundamental
set of solutions.
(iv) 𝑦 =
1 + 2 ln
( 𝑡
)
𝑡
