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Unformatted text preview: Solution of Ordinary Differential Equations James Keesling 1 General Theory Here we give a proof of the existence and uniqueness of a solution of ordinary differential equations satisfying certain conditions. The conditions are fairly minimal and usually satisfied for applications in physics and engineering. There are physical situations where the conditions are not satisfied. In those situations one may not be able to predict the path that the physical system will follow. The differential equation and initial value are given below. dx dt = f ( t,x ) x ( t ) = x Now suppose that f ( t,x ) is continuous on D = [ t- a,t + a ] [ x- b,x + b ] and that f ( t,x ) satisfies the Lipschitz condition | f ( t,x 2 ) = f ( t,x 1 ) | K | x 2- x 1 | on D for some K > 0. Under these hypotheses there is a positive d and a unique solution x ( t ) on the interval [ t- d,t + d ] satisfying x ( t ) = x ....
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- Summer '08