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Unformatted text preview: Lecture 10 Qualitative Behaviour of Solutions to ODEs The last few lectures were concerned with the explicit construction of solutions to general linear second order DEs. In many cases, a closed form solution is not possible and we have to resort to series solution methods. However, even if such a series solution is obtained, it is not obvious how such a solution behaves. For example, is it oscillatory? If so, can we get an idea of how quickly it oscillates? In applications, these are important questions. For this reason, it is useful to look at so-called qualitative methods that have been developed to analyze the qualitative behaviour of solutions, without actually determining the solutions explicitly. You may have already come in contact with some of these ideas in an earlier course in DEs. Well examine a first-order DE as a simple example of such qualitative analysis . Consider the following DE in y ( x ): dy dx = y (1 y ) , x . (1) You may recall that this DE, the so-called logistic equation is a simple model for population growth. For that reason, well focus on nonnegative solutions to (1). By looking at the RHS of this DE, we can come up with two solutions, without even formally solving the DE: 1. y ( x ) = 0 is a solution, and 2. y ( x ) = 1 is a solution. In both cases, the RHS is zero. The LHS will be zero since these are constant solutions. (This is one of the first things a DEs person will look for so-called equilibrium solutions.) Now lets see what happens when y is neither 0 nor 1: Case 1: Suppose that 0 < y ( x ) < 1 for an x 0. Then the RHS of (1) is positive, which implies that y ( x ) is increasing. But the graph of y ( x ) cannot cross the line y = 1, since this 1 would violate the Existence-Uniqueness theorem for first order DEs. (There is already the constant solution y ( x ) = 1 for all x . If the graph of y ( x ) crossed this line at, say, x 1 , then y ( x 1 ) would have two values.) These solutions will have to approach the value y = 1 from below as x . Case 2: Now suppose that y ( x ) > 1 for an x 0. Then the the RHS of (1) is negative, which implies that y ( x ) is decreasing. Once again, the graph of y ( x ) cannot cross the line y = 1, since this would violate the Existence-Uniqueness theorem for first order DEs. These solutions, then, will have to approach the value y = 1 from above as x . The above analysis leads to the following sketch of the qualitative behaviour of non-negative solutions to the logistic DE, for x > 0: 1 x y=1 y Let us now turn to second-order DEs and consider the following simple example: y + y = 0 . (2) Of course, we know that y 1 ( x ) = cos x and y 2 ( x ) = sin x are two linearly independent solutions to this DE, which implies that all nontrivial (i.e. nonzero) solutions are oscillatory. But this is not the spirit of qualitative analysis! Lets forget that we know these solutions and see what Eq. (2) can tell us about solutions. Amazingly, it can tell us a lot. One can actually deriveEq....
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This note was uploaded on 09/20/2010 for the course AMATH 351 taught by Professor Sivabalsivaloganathan during the Spring '08 term at Waterloo.
- Spring '08