Unformatted text preview: t 2 x 3 + 3 t4 x, we have ∂f ( t,x ) ∂t = 2 tx 3 +3 , here 4 x is a constant with respect to t , so the derivative is 0, and x 3 is also a constant, so t 2 x 3 when taking derivative against t we have 2 tx 3 . Therefore ﬁnding partial derivative is as easy as to ﬁnd ordinary derivative.This theorem only guarantees a solution deﬁned for t in an open interval, which might be a bounded interval. For example x = x 2 , we see that x ( t ) =1 t + c is the general solution, if c =2 then the solution is only deﬁned on interval (0, 2) as x ( t ) =1 t2 will become undeﬁned at t = 2 .Some time we could have more than one solution that satisﬁes the same initial condition, for example, for x = x 2 3 and x (0) = we have two diﬀerent solution, one is x ( t ) = 0 another one...
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 Spring '11
 Dr.Han
 Topology, Derivative, Continuous function

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