Existence and Uniqueness of System of Differential Equations

[cris(c)]

Hi everyone,

I'm not quite sure how to proceed to show existence (and perhaps uniqueness) of the following system of (first order) differential equations:
$\dot{x}=f(t_1,x,y,z)$
$\dot{y}=g(t_2,x,y,z)$
$\dot{z}=h(t_3,x,y,z)$

where $\dot{x}=\frac{\partial x}{\partial t_1}$, $\dot{y}=\frac{\partial y}{\partial t_2}$, and $\dot{z}=\frac{\partial z}{\partial t_3}$.

All existence theorems I've seen are formulated such that $t_1=t_2=t_3$. 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 $t_1 \neq t_2 \neq t_3$? Any help/reference where to look for such theorem would be greatly appreciate!

 

[lippyka]

Thanks!</code>Yes, the existence and uniqueness theorems can be applied to this system of differential equations even when t1, t2, and t3 are not equal. To prove this, you can use a theorem called the Picard–Lindelöf theorem. This theorem states that if f, g, and h are continuous functions and Lipschitz continuous with respect to x, y, and z then the solution to the system of equations exists and is unique.