1
$\begingroup$

I have the following problem:

Find the flow of the vector field $X=x\dfrac{\partial}{\partial x}+y(1+x)\dfrac{\partial}{\partial y}$ in $\mathbb{R}^2$.

I have tried to compute the Lie series for the second component but dont have any clue.

For the first component I have $(\Phi_t(x,y))_1=e^tx$. By integrating I got $e^{t+tx}y$ in the second component but that doesnt satisfy the definition of a flow ($\Phi_{s+t}=\Phi_s(\Phi_t)$).

Thanks in advance for any hint.

Regards, bronco

$\endgroup$
1
$\begingroup$

You have $x(t)=x_0 e^t$ and thus $\dot y = y (1 + x_0 e^t)$. So either $y=0$ identically, or $$ \dot y/y=1 + x_0 e^t \iff \ln|y|=t + x_0 e^t + C . $$ Thus, $$ y(t) = D e^{t+x_0 e^t} $$ (where $D=0$ or $D=\pm e^{C}$, respectively). Setting $t=0$ you get $D=y_0 e^{-x_0}$. Can you take it from there?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.