Solving a system of ODE with multiple 'time' variables

[cris(c)]

Homework Statement

Hi everyone,

Consider 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 $\dot{z}=\frac{\partial z}{\partial t_3}$.

Homework Equations

All existence theorems I know (picard and peano) are formulated such that $t_1=t_2=t_3$, but I'd like to know how to extend these results to the cae shown above.

The Attempt at a Solution

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!

 

[Antiphon]

I'm no ODE guru but it seems to me you need some type of relation between the three t variables.

 

[cris(c)]

Thanks for your answer. These variables are independent...what I want are conditions of
the $f$'s functions to ensure existence and uniqueness without having to solve the system...any idea how I should proceed?

I'm no ODE guru but it seems to me you need some type of relation between the three t variables.

 

[espen180]

Using different differentiation rules, and given a relations $t_2=t_2(t_1)$ and $t_3=t_3(t_1)$, you should be able to reduce the problem to a set of (possibly nonlinear nonautonomous) delay differential equations.

EDIT:
Sorry, I missed your last comment.

 

[cris(c)]

Using different differentiation rules, and given a relations $t_2=t_2(t_1)$ and $t_3=t_3(t_1)$, you should be able to reduce the problem to a set of (possibly nonlinear nonautonomous) delay differential equations.

EDIT:
Sorry, I missed your last comment.

No problem...should I infer that without this relation between the t variables I cannot proceed any further?

 

[espen180]

Not neccesarily. You should still be able to solve the system. Just keep in mind that since each f, g and h are implicitly dependent on all the t's, so all of x, y and z will also be dependent an all the t's. Also, you won't get neccesarily get constants as initial contitions, but functions. For example, the initial condition for x might be a function of t2 and t3.

 

[cris(c)]

Not neccesarily. You should still be able to solve the system. Just keep in mind that since each f, g and h are implicitly dependent on all the t's, so all of x, y and z will also be dependent an all the t's. Also, you won't get neccesarily get constants as initial contitions, but functions. For example, the initial condition for x might be a function of t2 and t3.

ok...understood. As initila conditions I have $f(0,x,y,z)=c_1$, $g(0,x,y,z)=c_2$, and $h(0,x,y,z)=c_1$, where $c$'s are constants. I don't want to actually solve the system, I want conditions on the f,g, and h functions that ensures existence (and uniqueness) of some solution. Is there any theorem that might help me?