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33bmidterm2

# 33bmidterm2 - Midterm 2 Math 33B Daniel Murfet Name Student...

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Midterm 2 Math 33B Daniel Murfet May 13, 2011 Name: Student ID: Discussion section: Signature: Q1: /3 Q2: /4 Q3: /4 Q4: /3 Q5: /4 Q6: /4 Total: /22 There are six problems and you have 50 minutes. There is some extra working paper at the end (if you use it, indicate this on the page with the question you are answering). You must show all work. 1

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Variation of parameters: Given a second order linear differential equation with forcing term f ( t ) and a fundamental set of solutions y 1 ( t ) , y 2 ( t ) of the associ- ated homogeneous equation, a particular solution to the inhomogeneous equation is given by v 1 y 1 + v 2 y 2 where v 1 = Z - y 2 f W , v 2 = Z y 1 f W , W = Wronskian of y 1 , y 2 . Q1 (3 points). Explain why the functions y 1 ( t ) = sin t and y 2 ( t ) = | sin t | are linearly independent on ( -∞ , ). 2
Q2 (4 points). The equation for an undamped forced spring-mass system is y 00 + ω 2 0 y = A sin ωt, (assume ω 0 6 = ω ) where y is the displacement from the spring-mass equilibrium, ω 0 is the natural frequency, ω is the forcing frequency and A is the amplitude of the forcing. (a) Derive the general solution to this differential equation. 3

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Q2 continues. (b) Find the solution with initial conditions

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33bmidterm2 - Midterm 2 Math 33B Daniel Murfet Name Student...

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