If parametric equations of a flow line are x = x(t), y = y(t), explain why these functions satisfy the differential equations dx/dt = x and dy/dt = −y.

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Step-by-step explanation:

The flowline vector field is given by F(x,y) = xî - yj

The velocity vector field of a flow's streamline is normally given as

V(x, y) = (dx/dt)î + (dy/dt)j

From the question, it is given that

(dx/dt) = x

(dy/dt) = -y

Hence, the velocity vector field for the flow's streamline in question is

V(x, y) = xî - yj

which corresponds or coincides with the flowline vector field equation of the flow.

The only time the pathline and streamline vector field coincide and have the same equation is when the flow is a steady state flow.

That is, the properties of the fluid flowing isn't changing with time!

Hope this Helps!!!

the radius is 2 times the diameter.

step-by-step explanation: