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I'm not quite sure how to proceed to show existence (and perhaps uniqueness) of the following system of (first order) differential equations:

[itex]\dot{x}=f(t_1,x,y,z) [/itex]

[itex]\dot{y}=g(t_2,x,y,z) [/itex]

[itex]\dot{z}=h(t_3,x,y,z) [/itex]

where [itex]\dot{x}=\frac{\partial x}{\partial t_1}[/itex], [itex]\dot{y}=\frac{\partial y}{\partial t_2}[/itex], and [itex]\dot{z}=\frac{\partial z}{\partial t_3}[/itex].

All existence theorems I've seen are formulated such that [itex]t_1=t_2=t_3[/itex]. I've tried reading the proofs to see if I can figure out a way to apply them to this problem, but I can't see how...Does someone knows whether these theorems hold true when [itex]t_1 \neq t_2 \neq t_3[/itex]? Any help/reference where to look for such theorem would be greatly appreciate!!!

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# System of ODE

Can you offer guidance or do you also need help?

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