The flow of a vector field

Question

Can somebody, please, help me? Let $X$ be a vector field on $\mathbb{R}^{2}$, $X=\frac{1}{x}\frac{\partial}{\partial x}+\frac{\partial}{\partial y}$. I was asked to calculate the flow of this vector field..I have to solve a system of differential equations: $\frac{dx}{dt}=\frac{1}{x}$ and $\frac{dy}{dt}=1$; we will have that $y(t)=t+c_{1}$, where $c_{1}$ is a real constant, but for the other one I will have two solutions: $x(t)=\sqrt{2(t+c_{2})}$ or $x(t)=-\sqrt{2(t+c_{2})}$, with $c_{2}$ also a real constant; what is the correct solution? Thank you very much!