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Mathematics LibreTexts

10.4E: Constant Coefficient Homogeneous Systems I (Exercises)

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Q10.4.1

In Exercises 10.4.1-10.4.15 find the general solution.

1. y=[1221]y

2. y=14[5335]y

3. y=15[43211]y

4. y=[1411]y

5. y=[2411]y

6. y=[4321]y

7. y=[6312]y

8. y=[112123411]y

9. y=[648404846]y

10. y=[358112111]y

11. y=[11212410617]y

12. y=[414432111]y

13. y=[226262222]y

14. y=[32227210105]y

15. y=[311351624]y

Q10.4.2

In Exercises 10.4.16-10.4.27 solve the initial value problem.

16. y=[7467]y,y(0)=[24]

17. y=16[7222]y,y(0)=[03]

18. y=[21122415]y,y(0)=[53]

19. y=[7467]y,y(0)=[17]

20. y=16[120410003]y,y(0)=[471]

21. y=13[223443210]y,y(0)=[115]

22. y=[638212335]y,y(0)=[011]

23. y=13[247155441]y,y(0)=[413]

24. y=[3011127103]y,y(0)=[276]

25. y=[251411453]y,y(0)=[8104]

26. y=[310420442]y,y(0)=[7102]

27. y=[226262222]y,y(0)=[6107]

Q10.4.3

28. Let A be an n×n constant matrix. Then Theorem 10.2.1 implies that the solutions of y=Ay

are all defined on (,).

  1. Use Theorem 10.2.1 to show that the only solution of (A) that can ever equal the zero vector is y0.
  2. Suppose y1 is a solution of (A) and y2 is defined by y2(t)=y1(tτ), where τ is an arbitrary real number. Show that y2 is also a solution of (A).
  3. Suppose y1 and y2 are solutions of (A) and there are real numbers t1 and t2 such that y1(t1)=y2(t2). Show that y2(t)=y1(tτ) for all t, where τ=t2t1.

Q10.4.4

In Exercises 10.4.29-10.4.34 describe and graph trajectories of the given system.

29. y=[1111]y

30. y=[43211]y

31. y=[93111]y

32. y=[11054]y

33. y=[54110]y

34. y=[7135]y

Q10.4.5

35. Suppose the eigenvalues of the 2×2 matrix A are λ=0 and μ0, with corresponding eigenvectors x1 and x2. Let L1 be the line through the origin parallel to x1.

  1. Show that every point on L1 is the trajectory of a constant solution of y=Ay.
  2. Show that the trajectories of nonconstant solutions of y=Ay are half-lines parallel to x2 and on either side of L1, and that the direction of motion along these trajectories is away from L1 if μ>0, or toward L1 if μ<0.

Q10.4.6

The matrices of the systems in Exercises 10.4.36-10.4.41 are singular. Describe and graph the trajectories of nonconstant solutions of the given systems.

36. y=[1111]y

37. y=[1326]y

38. y=[1313]y

39. y=[1212]y

40. y=[4411]y

41. y=[3131]y

Q10.4.6

42. Let P=P(t) and Q=Q(t) be the populations of two species at time t, and assume that each population would grow exponentially if the other didn’t exist; that is, in the absence of competition,

P=aPandQ=bQ,

where a and b are positive constants. One way to model the effect of competition is to assume that the growth rate per individual of each population is reduced by an amount proportional to the other population, so (A) is replaced by

P=aPαQQ=βP+bQ,

where α and β are positive constants. (Since negative population doesn’t make sense, this system holds only while P and Q are both positive.) Now suppose P(0)=P0>0 and Q(0)=Q0>0.

  1. For several choices of a, b, α, and β, verify experimentally (by graphing trajectories of (A) in the P-Q plane) that there’s a constant ρ>0 (depending upon a, b, α, and β) with the following properties:
    1. If Q0>ρP0, then P decreases monotonically to zero in finite time, during which Q remains positive.
    2. If Q0<ρP0, then Q decreases monotonically to zero in finite time, during which P remains positive.
  2. Conclude from (a) that exactly one of the species becomes extinct in finite time if Q0ρP0. Determine experimentally what happens if Q0=ρP0.
  3. Confirm your experimental results and determine γ by expressing the eigenvalues and associated eigenvectors of A=[ααβb]
    in terms of a, b, α, and β, and applying the geometric arguments developed at the end of this section.

This page titled 10.4E: Constant Coefficient Homogeneous Systems I (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform.

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