1.1: Problem Set
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For each of the ODEs below, write it as a first order system, state the dependent and independent variables, state any parameters in the ODE (i.e. unspecified constants) and state whether it is linear or nonlinear, and autonomous or nonautonomous,
(a)
\(\ddot{\theta}+\sigma\dot{\theta}+sin \theta = Fcos(\omega t)\), \(\theta \in \mathbb{S}^1\).
(b)
\(\ddot{\theta}+\sigma\dot{\theta}+\theta = Fcos(\omega t)\), \(\theta \in \mathbb{S}^1\).
(c)
\(\frac{d^{3}y}{dx^3}+x^{2}y\frac{dy}{dx}+y = 0\), \(x \in \mathbb{R}^1\).
(d)
\(\ddot{x}+\sigma\dot{x}+x-x^3 = \theta\),
\(\ddot{\theta} + sin \theta = 0\), \((x,\theta) \in \mathbb{R}^1 \times \mathbb{S}^1\)
(e)
\(\ddot{\theta}+\sigma\dot{\theta}+sin \theta = x\),
\(\ddot{x}-x+x^3 = 0\), \((\theta, x) \in \mathbb{R}^1 \times \mathbb{S}^1\)
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Consider the vector field:
\(\dot{x} = 3x^\frac{2}{3}\), \(x(0) \ne 0\), \(x \in \mathbb{R}\).
Does this vector field have unique solutions?
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Consider the vector field:
\(\dot{x} = -x+x^2\), \(x(0) = x_{0}\), \(x \in \mathbb{R}\).
Determine the time interval of existence of all solutions as a function of the initial condition, \(x_{0}\).
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Consider the vector field:
\(\dot{x} = a(t)x+b(t)\), \(x \in \mathbb{R}\).
Determine sufficient conditions on the coefficients a(t) and b(t) for which the solutions will exist for all time. Do the results depend on the initial condition?