Lebl-Basic-Analysis-I · Problem 6.3.3

The key insight is that Picard's theorem is about local behavior near a point, and shifting the initial condition doesn't change the underlying dynamics—it just moves where we're looking. Define a new variable $y = x - x_0$ to translate the problem so the initial point becomes the origin, apply the theorem you've already proved in that shifted coordinate system, and then translate the solution back. This shows that existence and uniqueness hold in a neighborhood of any starting point $x_0$, not just zero.

The key insight is that Picard's theorem is about local behavior near a point, and shifting the initial condition doesn't change the underlying dynamics—it just moves where we're looking. Define a new variable $y = x - x_0$ to translate the problem so the initial point becomes the origin, apply the theorem you've already proved in that shifted coordinate system, and then translate the solution back. This shows that existence and uniqueness hold in a neighborhood of any starting point $x_0$, not just zero.