Find flow of a given vector field

Question

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

Answer 1

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?