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- Homework Statement
- Find the flow of the vector field V(x, y) = (y, x)

- Relevant Equations
- V(x, y) = (y, x)

In part c, plotting the vector field shows the vector field is symmetric in x and y in the sets {x=y}.

in {x=y}, the variables can be interchanged and the solution becomes

x = x°e^t

y = y°e^tHowever, these solutions do not work for anywhere except {x=y} and don't satisfy dx/dt = y and dy/dt = x

yX+xY corresponds to the system of ODES

dx/dt = y

dy/dt = xWe eliminate dt from the above system to obtain the following solution to the system of ODES

(x^2)/2 - (y^2)/2=c

However I believe we cannot parametrize this equation to obtain the flow of the vector field.

What I did find from algebraic manipulations is

x = (x°e^t)/2 + (y°e^t)/2 + x°/2 - y°/2

y = (x°e^t)/2 + (y°e^t)/2 - x°/2 + y°/2

which satisfies the values t = 0, x = x°, y= y°

We tried to find the flow by t and s and the calculator tells us this is the wrong answer. Any ideas at all would be appreciated. thank you.