
#1
Jan513, 06:15 PM

P: 10

Given an open connected subset [itex]D[/itex] of the [itex](t,x)[/itex] plane and a function [itex]f\in C(D,\mathbb{R})[/itex], we say [itex]\varphi\in C^1(\text{proj}_1D,\mathbb{R})[/itex] is a solution of the first order differential equation [itex]x'=f(t,x)[/itex] if and only if [tex] \forall t\in \text{proj}_1D,\quad (t,\varphi(t))\in D
[/tex] and [tex]\forall t\in I, \quad \varphi'(t)=f(t,\varphi(t)) .[/tex] [itex]\textbf{Question}[/itex]: Is there a way to alter this definition so that the first condition after the 'iff' is automatically satisfied? Thanks in advance for any help. 


Register to reply 
Related Discussions  
Definition of Order of Operations  General Math  3  
Ohda!: Definition of Order in Baby Rudin  Calculus  2  
In a solution of 0.10 M H2SO4, the ions present in order of decreasing order.  Biology, Chemistry & Other Homework  1  
Confirm Precise Definition of a Limit solution  Calculus & Beyond Homework  2  
Firstorder, nonmodal definition of 'bag/multiset'  Set Theory, Logic, Probability, Statistics  7 