Question

I have the following system of differential equations:
\dot{x}=f(t_1,x,y,z)
\dot{y}=g(t_2,x,y,z)
\dot{z}=h(t_3,x,y,z)

Al existence theorems I know are formulated such that t_1=t_2=t_3. Reading the proof of these theorems doesn't help much if you want to see their validity when these variables aren't equal...Does someone knows whether these theorems hold true when t_1 \neq t_2 \neq t_3? Any help/reference where to look for such theorem would be greatly appreciate it!