Q10.4.1
In Exercises 10.4.1-10.4.15 find the general solution.
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Q10.4.2
In Exercises 10.4.16-10.4.27 solve the initial value problem.
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Q10.4.3
28. Let be an constant matrix. Then Theorem 10.2.1 implies that the solutions of
are all defined on .
- Use Theorem 10.2.1 to show that the only solution of (A) that can ever equal the zero vector is .
- Suppose is a solution of (A) and is defined by , where is an arbitrary real number. Show that is also a solution of (A).
- Suppose and are solutions of (A) and there are real numbers and such that . Show that for all , where .
Q10.4.4
In Exercises 10.4.29-10.4.34 describe and graph trajectories of the given system.
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Q10.4.5
35. Suppose the eigenvalues of the matrix are and , with corresponding eigenvectors and . Let be the line through the origin parallel to .
- Show that every point on is the trajectory of a constant solution of .
- Show that the trajectories of nonconstant solutions of are half-lines parallel to and on either side of , and that the direction of motion along these trajectories is away from if , or toward if .
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.
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Q10.4.6
42. Let and be the populations of two species at time , and assume that each population would grow exponentially if the other didn’t exist; that is, in the absence of competition,
where and 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
where and are positive constants. (Since negative population doesn’t make sense, this system holds only while and are both positive.) Now suppose and .
- For several choices of , , , and , verify experimentally (by graphing trajectories of (A) in the - plane) that there’s a constant (depending upon , , , and ) with the following properties:
- If , then decreases monotonically to zero in finite time, during which remains positive.
- If , then decreases monotonically to zero in finite time, during which remains positive.
- Conclude from (a) that exactly one of the species becomes extinct in finite time if . Determine experimentally what happens if .
- Confirm your experimental results and determine by expressing the eigenvalues and associated eigenvectors of in terms of , , , and , and applying the geometric arguments developed at the end of this section.