In that section we saw that all we needed to guarantee a unique solution was some basic continuity conditions. With boundary value problems we will often have no solution or infinitely many solutions even for very nice differential equations that would yield a unique solution if we had initial conditions instead of boundary conditions. In fact, a large part of the solution process there will be in dealing with the solution to the BVP.
Or maybe they will represent the location of ends of a vibrating string. So, the boundary conditions there will really be conditions on the boundary of some process. We know how to solve the differential equation and we know how to find the constants by applying the conditions.
We mentioned above that some boundary value problems can have no solutions or infinite solutions we had better do a couple of examples of those as well here.
This next set of examples will also show just how small of a change to the BVP it takes to move into these other possibilities. This, however, is not possible and so in this case have no solution. So, with Examples 2 and 3 we can see that only a small change to the boundary conditions, in relation to each other and to Example 1, can completely change the nature of the solution. All three of these examples used the same differential equation and yet a different set of initial conditions yielded, no solutions, one solution, or infinitely many solutions.
Note that this kind of behavior is not always unpredictable however. Also, note that with each of these we could tweak the boundary conditions a little to get any of the possible solution behaviors to show up i. All of the examples worked to this point have been nonhomogeneous because at least one of the boundary conditions have been non-zero. The solution is then,. Because of this we usually call this solution the trivial solution. Sometimes, as in the case of the last example the trivial solution is the only solution however we generally prefer solutions to be non-trivial.
This will be a major idea in the next section. Before we leave this section an important point needs to be made. In each of the examples, with one exception, the differential equation that we solved was in the form,.
The one exception to this still solved this differential equation except it was not a homogeneous differential equation and so we were still solving this basic differential equation in some manner. So, there are probably several natural questions that can arise at this point.
The answers to these questions are fairly simple. First, this differential equation is most definitely not the only one used in boundary value problems. It does however exhibit all of the behavior that we wanted to talk about here and has the added bonus of being very easy to solve. So, by using this differential equation almost exclusively we can see and discuss the important behavior that we need to discuss and frees us up from lots of potentially messy solution details and or messy solutions.
We will, on occasion, look at other differential equations in the rest of this chapter, but we will still be working almost exclusively with this one. There is another important reason for looking at this differential equation.
Admittedly they will have some simplifications in them, but they do come close to realistic problem in some cases.
As in Example 3 in the text, we solve these equations to find M and k 0. The initial condition P 0 implies that or ln 50 P P. The initial condition P 0 gives or ln P P. Our alligator population satisfies the equation dx 0. Thus the alligator population faces extinction in this event. Here we have the logistic equation dP 0. With P0 3. The initial condition M P C. Solving either of these for P yields the solution. Any way you look at it, you should conclude that the larger the parameter k 0 , the faster the logistic population P t approaches its limiting population M: To examine the question geometrically, we will assume that M 10 and that k1 1 and dP dP P 10 P and 2P 10 P.
We k2 2 , leading to the logistic equations dt dt draw slope fields and solution curves for each of these equations, using the same initial values P 0 in both cases:. These diagrams suggest that the larger the value of k, the more rapidly the population P t approaches the limiting population M.
For fixed M, t, and P0 this distance decreases as k increases; thus, the larger the value of k, the more rapidly P t approaches M. Finally, numerically, we tabulate values of P t , t 0, 0. Once again the evidence is that the larger value of k leads to the more rapid approach to M: k 1. Millions discover their favorite reads on issuu every month. Give your content the digital home it deserves.
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