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Chapter 5: Q23E (page 250)
In Problems 23 and 24, show that the given linear system is degenerate. In attempting to solve the system, determine whether it has no solutions or infinitely many solutions.
The given system has indefinitely many solutions.
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In Problems 3 鈥� 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.
In Problems 1鈥�7, convert the given initial value problem into an initial value problem for a system in normal form.
Use the result of Problem 31 to prove that all solutions to the equation\({\bf{y'' + }}{{\bf{y}}^{\bf{3}}}{\bf{ = 0}}\)remain bounded. [Hint: Argue that \(\frac{{{{\bf{y}}^{\bf{4}}}}}{{\bf{4}}}\) is bounded above by the constant appearing in Problem 31.]
Logistic Model.In Section 3.2 we discussed the logistic equation\(\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ = A}}{{\bf{p}}_{\bf{1}}}{\bf{p - A}}{{\bf{p}}^{\bf{2}}}{\bf{,p(0) = }}{{\bf{p}}_{\bf{o}}}\)and its use in modeling population growth. A more general model might involve the equation\(\frac{{{\bf{dp}}}}{{{\bf{dt}}}}{\bf{ = A}}{{\bf{p}}_{\bf{1}}}{\bf{p - A}}{{\bf{p}}^{\bf{r}}}{\bf{,p(0) = }}{{\bf{p}}_{\bf{o}}}\)where r>1. To see the effect of changing the parameter rin (25), take \({{\bf{p}}_{\bf{1}}}\)= 3, A= 1, and \({{\bf{p}}_{\bf{o}}}\)= 1. Then use a numerical scheme such as Runge鈥揔utta with h= 0.25 to approximate the solution to (25) on the interval\(0 \le {\bf{t}} \le 5\) for r= 1.5, 2, and 3What is the limiting population in each case? For r>1, determine a general formula for the limiting population.
Using the software, sketch the direction field in the phase-plane for the system . From the sketch, predict the asymptotic limit (as
of the solution starting at (1, 1).
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