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Math 250, section 511, Summer 1 2021

Direction Fields (Homework) INSTRUCTOR

Dianbin Bao Penn State Abington College

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5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Current Score

QUESTION

POINTS

TOTAL SCORE

4/33 12.1%

SAT, MAY 22, 2021 7:59 AM GMT+4

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Assignment Submission & Scoring

Assignment Submission

For this assignment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change the answer.

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Your last submission is used for your score.

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5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Write down a di!erential equation of the form whose solutions have the required behavior as

All solutions approach

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Write down a di!erential equation of the form whose solutions have the required behavior as

All solutions approach

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= ay + b dy dt

t → ∞.

y = 2. y' =

2−y

= ay + b dy dt

t → ∞.

y = . 7 8

y' =

78−y

1. [1/1 Points] BOYCEDIFFEQ10 1.1.007.DETAILS PREVIOUS ANSWERS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

2. [1/1 Points] BOYCEDIFFEQ10 1.1.008.DETAILS PREVIOUS ANSWERS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Write down a di!erential equation of the form whose solutions have the required behavior as

All other solutions diverge from

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Write down a di!erential equation of the form whose solutions have the required behavior as

All other solutions diverge from

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Consider the following di!erential equation. (A computer algebra system is recommended.)

= ay + b dy dt

t → ∞.

y = 3. y' =

y−3

= ay + b dy dt

t → ∞.

y = . 5 7

y' =

y−57

y' = y(5 − y)

3. [1/1 Points] BOYCEDIFFEQ10 1.1.009.DETAILS PREVIOUS ANSWERS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

4. [1/1 Points] BOYCEDIFFEQ10 1.1.010.DETAILS PREVIOUS ANSWERS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

5. [–/2 Points] BOYCEDIFFEQ10 1.1.011.DETAILS

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5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Draw a direction #eld for the given di!erential equation.

Based on the direction #eld, determine the behavior of y as If this behavior depends on the initial value of y at describe the dependency. Note that in this problem the equation is not of the form and the behavior of the solution is somewhat more complicated than for the equations in the text.

The equilibrium solutions are y(t) = 0 and y(t) = 5. The behavior of y(t) as t → ∞ depends on the initial value y(t0). If y(t0) > 0 then y(t) → 5 and if y(t0) < 0 then y(t) diverges from y = 0.

The equilibrium solutions are y(t) = 0 and y(t) = 5. The behavior of y(t) as t → ∞ depends on the initial value y(t0). If y(t0) > 5 then y(t)

diverges from y = 5. If 0 < y(t0) < 5 then y(t) → 5. If y(t0) < 0 then y(t) diverges from y = 0.

The equilibrium solution is y(t) = −5. The behavior of y(t) as t → ∞ is independent of the initial value y(t0), so y(t) → −5 for all y(t0).

t → ∞. t = 0, y' = ay + b,

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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The equilibrium solutions are y(t) = 0 and y(t) = −5. The behavior of y(t) as t → ∞ depends on the initial value y(t0). If y(t0) > 0 then y(t) → −5 and if y(t0) < 0 then y(t) diverges from y = 0.

The equilibrium solution is y(t) = 5. The behavior of y(t) as t → ∞ is independent of the initial value y(t0), so y(t) → 5 for all y(t0).

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Consider the following di!erential equation. (A computer algebra system is recommended.)

Draw a direction #eld for the given di!erential equation. y' = y(y − 5)2

6. [–/2 Points] BOYCEDIFFEQ10 1.1.014.DETAILS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Based on the direction #eld, determine the behavior of y as If this behavior depends on the initial value of y at describe the dependency. Note that in this problem the equation is not of the form and the behavior of the solution is somewhat more complicated than for the equations in the text.

The equilibrium solutions are y(t) = 0 and y(t) = 5. The behavior of y(t) as t → ∞ depends on the initial value y(t0). If y(t0) > 5 then y(t)

diverges from y = 5. If 0 < y(t0) < 5 then y(t) → 5. If y(t0) < 0 then y(t) diverges from y = 0.

The equilibrium solutions are y(t) = 0 and y(t) = −5. The behavior of y(t) as t → ∞ depends on the initial value y(t0). If y(t0) > 0 then y(t) → −5 and if y(t0) < 0 then y(t) diverges from y = 0.

The equilibrium solution is y(t) = −5. The behavior of y(t) as t → ∞ is independent of the initial value y(t0), so y(t) → −5 for all y(t0).

The equilibrium solutions are y(t) = 0 and y(t) = 5. The behavior of y(t) as t → ∞ depends on the initial value y(t0). If y(t0) > 0 then y(t) → 5 and if y(t0) < 0 then y(t) diverges from y = 0.

The equilibrium solution is y(t) = 5. The behavior of y(t) as t → ∞ is independent of the initial value y(t0), so y(t) → 5 for all y(t0).

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t → ∞. t = 0, y' = ay + b,

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Consider the following list of di!erential equations, one of which produced the direction #eld shown below. Identify the di!erential equation that corresponds to the given direction #eld.

(a) y' = 2y − 1

(b) y' = 2 + y

(c) y' = y − 2

(d) y' = y(y + 1)

(e) y' = y(y − 1)

(f) y' = 1 + 2y

(g) y' = −2 − y

(h) y' = y(1 − y)

(i) y' = 1 − 2y

(j) y' = 2 − y

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7. [–/1 Points] BOYCEDIFFEQ10 1.1.015.DETAILS

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5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Consider the following list of di!erential equations, one of which produced the direction #eld shown below. Identify the di!erential equation that corresponds to the given direction #eld.

y' = 2y − 1

y' = 2 + y

y' = y − 2

y' = y(y + 1)

y' = y(y − 1)

y' = 1 + 2y

y' = −2 − y

y' = y(1 − y)

y' = 1 − 2y

y' = 2 − y

GO Tutorial

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8. [–/1 Points] BOYCEDIFFEQ10 1.1.015.GO.DETAILS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Consider the following list of di!erential equations, one of which produced the direction #eld shown below. Identify the di!erential equation that corresponds to the given direction #eld.

y' = 2y − 1

y' = 2 + y

y' = y − 2

y' = y(y + 3)

y' = y(y − 3)

y' = 1 + 2y

y' = −2 − y

y' = y(3 − y)

y' = 1 − 2y

y' = 2 − y

GO Tutorial

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9. [–/1 Points] BOYCEDIFFEQ10 1.1.016.GO.DETAILS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Consider the following list of di!erential equations, one of which produced the direction #eld shown below. Identify the di!erential equation that corresponds to the given direction #eld.

(a) y' = 5y − 1

(b) y' = 5 + y

(c) y' = y − 5

(d) y' = y(y + 6)

(e) y' = y(y − 6)

(f) y' = 1 + 5y

(g) y' = −5 − y

(h) y' = y(6 − y)

(i) y' = 1 − 5y

(j) y' = 5 − y

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10. [–/1 Points] BOYCEDIFFEQ10 1.1.017.DETAILS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Consider the following list of di!erential equations, one of which produced the direction #eld shown below. Identify the di!erential equation that corresponds to the given direction #eld.

y' = 2y − 1

y' = 2 + y

y' = y − 2

y' = y(y + 3)

y' = y(y − 3)

y' = 1 + 2y

y' = −2 − y

y' = y(3 − y)

y' = 1 − 2y

y' = 2 − y

GO Tutorial

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11. [–/1 Points] BOYCEDIFFEQ10 1.1.017.GO.DETAILS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Consider the following list of di!erential equations, one of which produced the direction #eld shown below. Identify the di!erential equation that corresponds to the given direction #eld.

y' = 2y − 1

y' = 2 + y

y' = y − 2

y' = y(y + 1)

y' = y(y − 1)

y' = 1 + 2y

y' = −2 − y

y' = y(1 − y)

y' = 1 − 2y

y' = 2 − y

GO Tutorial

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12. [–/1 Points] BOYCEDIFFEQ10 1.1.018.GO.DETAILS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Consider the following list of di!erential equations, one of which produced the direction #eld shown below. Identify the di!erential equation that corresponds to the given direction #eld.

(a) y' = 4y − 1

(b) y' = 4 + y

(c) y' = y − 4

(d) y' = y(y + 3)

(e) y' = y(y − 3)

(f) y' = 1 + 4y

(g) y' = −4 − y

(h) y' = y(3 − y)

(i) y' = 1 − 4y

(j) y' = 4 − y

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13. [–/1 Points] BOYCEDIFFEQ10 1.1.019.DETAILS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Consider the following list of di!erential equations, one of which produced the direction #eld shown below. Identify the di!erential equation that corresponds to the given direction #eld.

y' = 8y − 1

y' = 8 + y

y' = y − 8

y' = y(y + 7)

y' = y(y − 7)

y' = 1 + 8y

y' = −8 − y

y' = y(7 − y)

y' = 1 − 8y

y' = 8 − y

GO Tutorial

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14. [–/1 Points] BOYCEDIFFEQ10 1.1.019.GO.DETAILS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Consider the following list of di!erential equations, one of which produced the direction #eld shown below. Identify the di!erential equation that corresponds to the given direction #eld.

y' = 6y − 1

y' = 6 + y

y' = y − 6

y' = y(y + 7)

y' = y(y − 7)

y' = 1 + 6y

y' = −6 − y

y' = y(7 − y)

y' = 1 − 6y

y' = 6 − y

GO Tutorial

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15. [–/1 Points] BOYCEDIFFEQ10 1.1.020.GO.DETAILS

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5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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A pond initially contains 1,000,000 gal of water and an unknown amount of an undesirable chemical. Water containing 0.06 g of this chemical per gallon $ows into the pond at a rate of 400 gal/hr. The mixture $ows out at the same rate, so the amount of water in the pond remains constant. Assume that the chemical is uniformly distributed throughout the pond.

(a) Write a di!erential equation for the amount of chemical in the pond at any time. (Let q denote the amount of chemical in the pond at time

(b) How much of the chemical will be in the pond after a very long time?

grams

Does this limiting amount depend on the amount that was present initially?

The limiting amount ---Select--- depend on the amount that was present initially.

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A spherical raindrop evaporates at a rate proportional to its surface area. Write a di!erential equation for the volume V of the raindrop as a function of time. (Use k for the constant of proportionality.)

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t.)

=

dq dt

grams hour

=

for k > 0

dV dt

16. [–/3 Points] BOYCEDIFFEQ10 1.1.021.DETAILS

MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER

17. [–/1 Points] BOYCEDIFFEQ10 1.1.022.DETAILS MY NOTES ASK YOUR TEACHER

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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A certain drug is being administered intravenously to a hospital patient. Fluid containing 3 mg/cm3 of the drug enters the patient's bloodstream

at a rate of 200 cm3/h. The drug is absorbed by body tissues or otherwise leaves the bloodstream at a rate proportional to the amount present,

with a rate constant of

(a) Assuming that the drug is always uniformly distributed throughout the bloodstream, write a di!erential equation for the amount of drug that is present in the bloodstream at any time. (Let M be the total amount of the drug (in milligrams) in the patient's body at any given time t in hours.)

(b) How much of the drug is present in the bloodstream after a long time?

mg

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Consider the following di!erential equation. (A computer algebra system is recommended.)

Draw a direction #eld for the given di!erential equation.

0.5 (h)−1.

=

dM dt

mg h

y' = −1 + t − y

18. [–/2 Points] BOYCEDIFFEQ10 1.1.024.DETAILS

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19. [–/2 Points] BOYCEDIFFEQ10 1.1.026.DETAILS

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5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Based on the direction #eld, determine the behavior of y as If this behavior depends on the initial value of y at describe this dependency. Note that the right side of this equation depends on t as well as y; therefore, its solution can exhibit more complicated behavior than those in the text.

t → ∞. t = 0,

The behavior of y(t) is independent of the initial value y(t0): y(t) → 0 for all y(t0).

The behavior of y(t) is independent of the initial value y(t0): y(t) → t − 2 for all y(t0).

Depending on the initial value y(t0), either y(t) diverges from y = − sin t + − 1 or the solution is y(t) = − sin t + − 1. 2

2 ! 4

2

2 ! 4

The behavior of y(t) is independent of the initial value y(t0) and diverges from y = t − 2 for all y(t0).

The behavior of y(t) is independent of the initial value y(t0) and diverges from y = 0 for all y(t0).

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Consider the following di!erential equation. (A computer algebra system is recommended.)

Draw a direction #eld for the given di!erential equation.

y' = e−t + y

20. [–/2 Points] BOYCEDIFFEQ10 1.1.028.DETAILS MY NOTES ASK YOUR TEACHER

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

Page 21 of 27https://www.webassign.net/web/Student/Assignment-Responses/last?dep=26727275

Based on the direction #eld, determine the behavior of y as If this behavior depends on the initial value of y at describe this dependency. Note that the right side of this equation depends on t as well as y; therefore, its solution can exhibit more complicated behavior than those in the text.

Additional Materials

eBook

Consider the following di!erential equation. (A computer algebra system is recommended.)

Draw a direction #eld for the given di!erential equation.

t → ∞. t = 0,

The behavior of y(t) is independent of the initial value y(t0) and diverges from y = 0 for all y(t0).

Depending on the initial value y(t0), either y(t) → −∞ or y(t) → .2t − 1

The behavior of y(t) is independent of the initial value y(t0) and diverges from y = t − 2 for all y(t0).

The behavior of y(t) is independent of the initial value y(t0): y(t) → 0 for all y(t0).

The behavior of y(t) is independent of the initial value y(t0): y(t) → t − 2 for all y(t0).

y' = 2 sin t + 1 + y

21. [–/2 Points] BOYCEDIFFEQ10 1.1.030.DETAILS

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5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

Page 22 of 27https://www.webassign.net/web/Student/Assignment-Responses/last?dep=26727275

Based on the direction #eld, determine the behavior of y as If this behavior depends on the initial value of y at describe this dependency. Note that the right side of this equation depends on t as well as y; therefore, its solution can exhibit more complicated behavior than those in the text.

Additional Materials

eBook

Consider the following di!erential equation. (A computer algebra system is recommended.)

Draw a direction #eld for the given di!erential equation.

t → ∞. t = 0,

Depending on the initial value y(t0), either y(t) diverges from y = − sin t + − 1 or the solution is y(t) = − sin t + − 1. 2

2 ! 4

2

2 ! 4

The behavior of y(t) is independent of the initial value y(t0): y(t) → t − 2 for all y(t0).

The behavior of y(t) is independent of the initial value y(t0) and diverges from y = t − 2 for all y(t0).

The behavior of y(t) is independent of the initial value y(t0): y(t) → 0 for all y(t0).

The behavior of y(t) is independent of the initial value y(t0) and diverges from y = 0 for all y(t0).

y' = 5t − 1 − y2

22. [–/2 Points] BOYCEDIFFEQ10 1.1.031.DETAILS

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5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Based on the direction #eld, determine the behavior of y as If this behavior depends on the initial value of y at describe this dependency. Note that the right side of this equation depends on t as well as y; therefore, its solution can exhibit more complicated behavior than those in the text.

t → ∞. t = 0,

The behavior of y(t) is independent of the initial value y(t0): y(t) → 0 for all y(t0).

The behavior of y(t) is independent of the initial value y(t0) and diverges from y = t − 5 for all y(t0).

Depending on the initial value y(t0), either y(t) → −∞ or y(t) → .5t − 1

The behavior of y(t) is independent of the initial value y(t0) and diverges from y = 0 for all y(t0).

The behavior of y(t) is independent of the initial value y(t0): y(t) → t − 5 for all y(t0).

5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Consider the following di!erential equation. (A computer algebra system is recommended.)

Draw a direction #eld for the given di!erential equation.

y' = −(7t + y)

7y

23. [–/2 Points] BOYCEDIFFEQ10 1.1.032.DETAILS

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5/19/21, 11:17 AMDirection Fields - Math 250, section 511, Summer 1 2021 | WebAssign

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Based on the direction #eld, determine the behavior of y as If this behavior depends on the initial value of y at describe this dependency. Note that the right side of this equation depends on t as well as y; therefore, its solution can exhibit more complicated behavior than those in the text.

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t → ∞. t = 0,

The behavior of y(t) is independent of the initial value y(t0) and diverges from y = 0 for all y(t0).

Depending on the initial value y(t0), either y(t) → −∞ or y(t) → .2t − 1

The behavior of y(t) is independent of the initial value y(t0): y(t) → 0 for all y(t0).

Depending on the initial value y(t0), either y(t) → ∞ or y(t) → 0.

The behavior of y(t) is independent of the initial value y(t0): y(t) → t for all y(t0).

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