- #1
cris(c)
- 26
- 0
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:
[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!
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!