I had read the following in a paper:
Let $M = \mathbb R^3$ be a differential manifold, $X$ be a vector field with $X(x,y,z) = \frac{\partial}{\partial x} + x\frac{\partial}{\partial z} = [1,\, 0,\, x]^t$, then we get a 1-parameter group of diffeomorphism starting from $(x, y, z)$ as $\varphi_s(x, y, z) = [x+s,\, y,\, z+xs+\frac{1}{2}s^2]^t$.
How is it a 1-parameter group of $X$? Shouldn't it be $\varphi_s(x, y, z) = [x+s,\, y,\, z+xs]^t$ ?
Written out explicitly, you have to solve the system of ODEs given by $x'(s)=1$, $y'(s)=0$, $z'(s)=x(s)s$ with initial conditions $x(0)=x$, $y(0)=y$ and $z(0)=z$. So you readily get $y(s)=y$ and $x(s)=x+s$, so you have to solve $z'(s)=x+s$ and the solution indeed is $z(s)=z+xs+\frac12 s^2$ as claimed in the paper.