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B.1 Existence and uniqueness of solutions There are functions f:RR for which the solution x() of the differential equation is not unique: Example B.1.1 (Non-unique solution). A standard example of a differential equation with a non-unique solution is i(t) = V(x(t), x(0) = 0, t 0. Clearly the zero function x() 0V is one solution, but it is easy to verify that for every c> the function 0 te (0, c] x(t) x(t) = (t-c)2 >c 0 129 is a solution as well! Weird. It is as if the state x(t) - like Baron Munchhausen is able to lift The vector field function in this example is f(x) - v/x and it has unbounded derivative around x = O. We will see next that if the function f(x) does not increase too quickly then uniqueness is ensured. A measure for the rate of increase is the Lipschitz constant.

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