$($a$)$ Sketch the vector field $F(x,y)=i+7xj$ and then sketch some flow lines. What shape do these flow lines appear to have?

parabolas

rays

ellipses

hyperbolas

straight lines

$($b$)$ If parametric equations for the flow lines are $x=x(t),y=y(t),$ what differential equations do these functions satisfy?

$\frac{dx}{dt}=$

$\frac{dy}{dt}=$

Deduce that $\frac{dy}{dx}=7x.$

$\frac{dy}{dx}=\frac{d\frac{y}{d}t}{d\frac{x}{d}t}=7x$

$\frac{dy}{dx}=\frac{d\frac{x}{d}t}{d\frac{y}{d}t}=7x$

$\frac{dy}{dx}=\frac{d\frac{t}{d}y}{d\frac{x}{d}t}=7x$

$\frac{dy}{dx}=\frac{d\frac{y}{d}t}{d\frac{t}{d}x}=7x$

$\frac{dy}{dx}=\frac{d\frac{t}{d}x}{d\frac{y}{d}t}=7x$

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