vector field v=−4xi−4yj (a) find the system of differential equations associated with the vector field: x′= y′= (b) Solve the system you found above to find the flow x(t)= y(t)=

Answer 1

I assume that we are suppose to interpret x, y as coordinates of position, and

v as the velocity vector.

If my assumption is correct then v is defined as

v = x'i + y'j

So

x'i + y'j = -4xi - 4yj

Therefore

x' = -4x

y' = -4y

That is the answer the question (a)

For question (b) use separation of variables.

dx/dt = -4x

∫dx/x = ∫-4dt

ln(x) = -4t + C

x = e^-4te^C

Therefore the solution to the first equation is of the form

x(t) = Ae^-4t for some constant A.

Analogously, the solution to the second equation is

y(t) = Be^-4t for some constant B.

The constants A and B would be determined from initial conditions, if given.