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nonlinear_example - An Example of a Non-linear Dierential...

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An Example of a Non-linear Differential Equation R. C. Daileda In class we mentioned the following theorem, whose proof the interested reader can find in Section 2.8 of [1]. Theorem. 1. Consider the initial value problem (IVP) y = f ( t, y ) , y ( t 0 ) = y 0 . (1) If f and ∂f/∂y are both continuous on a disk centered at ( t 0 , y 0 ) then (1) has a unique solution defined on some interval t 0 h < t < t 0 + h , h > 0 . Several points should be made here. While this theorem does give us an effective way of determining if a solution to an IVP exists, it gives us no way of determining this solution nor does it give any information on the interval of definition of that solution. This should be compared to the analogous fact for IVPs involving linear first order ODEs (Theorem 2.4.1 of [1]). If f is linear then we can write down an explicit solution, and the interval of definition of that solution can be determined from f and t 0 alone . The moral is that the behavior of solutions to non-linear differential equations can be drastically different than that of linear equations, as the following example is meant to illustrate.
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