Is the Solution to dx/dt=f(x) with x(0)=xo Unique?

[l888l888l888]

Homework Statement

Show that the solution of dx/dt=f(x), x(0)=xo, f in C^1(R), is unique

Homework Equations

C^1(R) is the set of all functions whose first derivative is continous.
F(x)=integral from xo and x (dy/f(y))

The Attempt at a Solution

Assume phi1(x) and phi2(x) are both soultions. Then d(phi1(x))/dt=f(phi1(x)) and d(phi2(x))/dt=f(phi2(x)). Consider phi1(x)-phi2(x). d(phi1(x)-phi2(x))/dt= f(phi1(x))-f(phi2(x))...
I need to prove the two solutions are infact equal. Also it says in my book that every solution x(t) must satisfy F(x(t))=t, with phi(t)=F^-1 (t) (F Inverse)

 

[lanedance]

shouldn't the two solutions be functions of t?

 

[l888l888l888]

yes. I am sorry. I made a mistake. the two solutions hsould be phi1(t) and phi2(t).