APPM 4/5350 Worksheet Week 1 Figure 1: www.xkcd.com 1. ODEs (a) Find a solution θ ( t ) for: m ¨ θ =-g sin( θ ) with small angle approximation sin( θ ) ≈ θ , and initial conditions θ (0) = 0 , ˙ θ (0) = 1 (b) Obtain a general solution y ( t ) (including arbitrary constants) for y0 + βty = 1 β = const 2. Equilibrium Solution Find the general equilibrium solution (including arbitrary constants) for: ∂ 2 g ∂t 2 = ∂ 2 g ∂x 2-αg, α >0 What does it mean to have an equilibrium solution and given g ( x,t ) how would one ﬁnd the equilibrium behavior if it exists.
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