# Find flow of a given vector field

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

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?