Id: 2327-f2015-1.m4,v 1.2 2015/10/04 19:49:34 john Exp john
1. Solve the initial value problem x ( t
0
) = x
0 for the differential equation x
′ ( t ) = − p + qx ( t ) − rx ( t ) 2 in the exceptional case q
2 = 4 pr . What is the largest interval on which wyour solution is defined? What is the largest interval on which it is biologically meaningful, with the usual interpretation of x as the population of a species? Don’t forget to check that the solution you find really is a solution and that it’s the only solution.
2. Show that if q
2
< 4 pr then x
′ ( t ) = − p + qx ( t ) − rx ( t ) 2 has no solutions which extend to t = ∞ and remain positive, assuming, as usual, that p, q, r > 0. Biologically this means there are no solutions which do not lead sooner or later to extinction of the species.
Note: You can do this by solving the equation explicitly, but please don’t. Instead show that x ′ ( t ) ≤ −
4 pr − q
2
4 r
< 0 and then use that inequality. You will need the Mean Value Theorem, or something equivalent.
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