Except for the case of constant solutions, it is usually not possible to find explicit expression for the solutions of a nonlinear autonomous system, However, if the system contains such an expression as x^2+y^2 we can solve some nonlinear systems by changing to polar coordinates. (a) Let x =r cos θ and y = r sin θ where r and θ are functions of t. Express dr/dt and dθ/dt with respect to x, y, dx/dt,dy/dt, and r. b) Consider the following nonlinear plane autonomous system dx/dt = -y-x√x^2+y^2 dy/dt= x-y√x^2+y^2 satisfying the initial condition x(0)=3 and y(0) =3. Express the system (DE1) in polar coordinates. c) Solve the system you rewrote in part (b) for r and θ.
Except for the case of constant solutions, it is usually not possible to find explicit expression for the
solutions of a nonlinear autonomous system, However, if the system contains such an expression as x^2+y^2
we can solve some nonlinear systems by changing to polar coordinates.
(a) Let x =r cos θ and y = r sin θ where r and θ are functions of t. Express dr/dt and dθ/dt with respect to x, y, dx/dt,dy/dt, and r.
b) Consider the following nonlinear plane autonomous system
dx/dt = -y-x√x^2+y^2
dy/dt= x-y√x^2+y^2
satisfying the initial condition x(0)=3 and y(0) =3. Express the system (DE1) in polar coordinates.
c) Solve the system you rewrote in part (b) for r and θ.
d) Sketch the trajectory in part (c).
e) Identify and classify the critical point of the system (DE1).
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