 Solution

Follow a fluid particle released at time $s$, and use $\tau$ for its running time. Its pathline satisfies $dx/d\tau =1$, $dy/d\tau =\tau x$ and $dz/d\tau =0$, with zero position at $\tau =s$. Hence $x(\tau )=\tau -s$, $z(\tau )=0$, andAt observation time $t$, set $x=t-s$. Substitution gives the streaklineThe curve is parameterized by release times $s\le t$. If release starts at time zero, only $0\le x\le t$ is occupied. This is a streakline, the positions of different particles released at one fixed point, rather than an instantaneous streamline of this unsteady flow.