MATH 316, DIFFERENTIAL EQUATIONS, WINTER 2000

Problem Set 10, due Friday, April 14

Problem 1 Find the location and type of all equilibrium points and draw a phase portrait of the following non-linear differential equations:

\begin{aligned}(a) & x′ = y(x^{2} − 4) \\ & y′ = x(1 − y^{2}) \end{aligned}  \begin{aligned}(b) & x′ = xy \\ & y′ = 1 − x^{2} − y^{2} \end{aligned}  \begin{aligned}(c) & x′ = sin(3x + y) \\ & y′ = 3 sin(\frac{x}{3} + y) \end{aligned}

Problem 2 Consider the following simple spring pendulum:

The mass is m, and the spring constant is k. The relaxed length of the spring is l_{0}. This is a 2-dimensional system. Let q_{0} = α be the angle between the pendulum and a vertical line, as in the illustration, and let q_{1} = l be the length of the arm of the pendulum.

(a) Find formulas for the potential and kinetic energy in terms of q_{0}, q_{1}, q̇_{0}, and q̇_{1}.

(b) Use the Lagrangian formula to find the equation of motion for this system.