(=] [=] [=] [#] Phase plane diagrams and first-order exact equations Consider the two-dimensional velocity field of a fluid flow given by u = (u(z,y),v(z,y)) = (* siny, 3z%y). Let & = (x(t), y()) describe the path of a fluid particle as it moves through the fluid. The motion of the particle can be modelled with the following system of first-order ordinary differential equations dy o 9 dtv 3zy (1) dz 3 I - u= siny. 1. Determine the equations for the nullclines of the system. Hint: It may be convenient to express some of the nullclines in implicit form f(z,y) = C, where C' is a constant. . Find the equilibrium points of the system. . Use the chain rule to derive a first-order ODE for the trajectories in the xy phase plane. . Re-write the ODE in 3 in differential form. T e W N . Use the solution method for exact equations, including the test for exactness, to find the general solution to the ODE in 4