$($a$)$ Sketch the vector field

F $=$ i $+3$xj

and then sketch some flow lines. What shape do these flow lines appear to have?

parabolasstraight lines$$hyperbolasraysellipses

$($b$)$ If parametric equations for the flow lines are

x $=$ x$($t$),$ y $=$ y$($t$),$

what differential equations do these functions satisfy? Deduce that

dy$/$dx $=3$x$.$

$($c$)$ If a particle starts at the origin in the velocity field given by F$,$ find an equation of the path it follows.