MHB Existence of Unique Solution for Nonlinear System with Arbitrary Constants

  • Thread starter Thread starter Julio1
  • Start date Start date
  • Tags Tags
    Nonlinear System
Julio1
Messages
66
Reaction score
0
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.
 
Physics news on Phys.org
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?
 
There is the following linear Volterra equation of the second kind $$ y(x)+\int_{0}^{x} K(x-s) y(s)\,{\rm d}s = 1 $$ with kernel $$ K(x-s) = 1 - 4 \sum_{n=1}^{\infty} \dfrac{1}{\lambda_n^2} e^{-\beta \lambda_n^2 (x-s)} $$ where $y(0)=1$, $\beta>0$ and $\lambda_n$ is the $n$-th positive root of the equation $J_0(x)=0$ (here $n$ is a natural number that numbers these positive roots in the order of increasing their values), $J_0(x)$ is the Bessel function of the first kind of zero order. I...
Are there any good visualization tutorials, written or video, that show graphically how separation of variables works? I particularly have the time-independent Schrodinger Equation in mind. There are hundreds of demonstrations out there which essentially distill to copies of one another. However I am trying to visualize in my mind how this process looks graphically - for example plotting t on one axis and x on the other for f(x,t). I have seen other good visual representations of...
Back
Top