Existence of Unique Solution for Nonlinear System with Arbitrary Constants

[Julio1]

Show that the nonlinear system

$\dot{X_1}=2\cos X_2, X_1(0)=a$

$\dot{X_2}=3\sin X_1, X_2(0)=b$

has a unique solution for the arbitrary constants $a$ and $b$.

how to solve this system? Thanks.

 

[S.G. Janssens]

You are asked to show that a unique solution exists, but you don't need to necessarily find an expression for it. Write
\[
\dot{x}(t) = f(x(t)), \qquad t \in \mathbb{R}, \qquad x(0) = (a,b),
\]
with $f : \mathbb{R}^2 \to \mathbb{R}^2$ given by $f(x) = (2\cos{x_2}, 3\sin{x_1})$. Do you know a theorem that relates the Lipschitz continuity of $f$ to the existence of a unique solution to the above initial-value problem?