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Extra info for A Course in Applied Mathematics, Vols 1 & 2
However, the results which we obtained were radically different because of the second boundary condition. These examples illustrate the complexities inherent in the study of boundary value problems. These complexities can, at least partially, be attributed to the interaction of the boundary conditions with the differential equation. Because of these inherent complexities, we shall not present any general theory for boundary value problems. 3 2 1. Verify that y(x) = c1ex is a one-parameter family of solutions of the differential equation y − 2xy = 0 on the interval (−∞, ∞).
Thus, y(x) = cx is a solution of the IVP (13) on the interval (−∞, ∞) for any choice of the constant c. Consequently, the IVP (13) is an example of an initial value problem with an infinite number of solutions. EXAMPLE 5 An Initial Value Problem with a Unique Solution Now consider the initial value problem (14) xy − y = 0; y(1) = 4. Again, the function y(x) = cx, where c is an arbitrary constant, is a oneparameter family of solutions of the differential equation xy − y = 0 on the interval (−∞, ∞).
In geometric terms, there is no solution of the differential equation xy −y = 0 which passes through any point on the positive or negative y-axis—that is, any point of the form (0, d) where d = 0. © 2010 by Taylor and Francis Group, LLC 24 Ordinary Differential Equations EXAMPLE 4 An Initial Value Problem with an Infinite Number of Solutions The differential equation of the initial value problem (13) xy − y = 0; y(0) = 0 is the same as the differential equation in the previous example. As we noted earlier, y = cx is a one-parameter family of solutions on (−∞, ∞).