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Solve the nonlinear system

Julio

Member
Feb 14, 2014
71
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.
 

Janssens

Well-known member
Sep 16, 2017
202
The Netherlands
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?