� � � � � � 18.034 PRACTICE EXAM 2, SPRING 2004 Problem 1 Let r be a positive real number. Consider the 2nd order, linear differential equation, 3 2r 3 y�� − r + y + + y = 2 0, t t ttert where y(t) is a function on (0, ∞). One solution of this equation is y1(t) = . Use Wronskian reduction of order to find a second solution y2(t). Problem 2 An undamped harmonic oscillator satisfies the ODE, 2 y�� + ω y = 0. Let y(t) be a solution of this ODE for t < τ. At some time τ > 0, the oscillator is given an impulse of size v > 0. In other words, if limt τ− y(t) = y0, limt→τ− y�(t) = v0 →then for t > τ, y(t) is a solution of the IVP, ⎧⎨ ⎩ y�� + ω2y = 0, y(τ) = y0, y�(τ) = v0 + v (a) Write y(t) in normal form A cos(ωt − φ) for t < τ , and in normal form y(t) = B cos(ωt − ψ) for t > τ. Find an equation expressing B2 in terms of A2 , v0 and v. (b) If the goal of the impulse is to maximize the amplitude B, at what moment τ in the cycle of the oscillator should the impulse be applied? If the goal is minimize the amplitude B, at what moment τ should the impulse be applied? Problem 3 Consider the following constant coefficient linear ODE, y��� + y = 0. (a) Find the characteristic polynomial and find all re al and complex roots. (b) Find the general real-valued solution of the ODE. (c) Find a particular solution of the driven ODE, y��� + y = cos(√3t/2). Problem 4 The linear ODE, y�� + (t − 3/t)y� − 2y = 0, 2thas a basic solution pair y1(t) = e− 2 , y2(t) = t2 − 2. (a) Find the Wronskian W [y1, y2](t). (b) Use variation of parameters to find a particular solution of the driven ODE, 4 y�� + (t − 3/t)y� − 2y = t . Date : Spring 2004. 1� � Problem 5 Re call that PCR(0, 1] is the set of all piecewise continuous real-valued functions on the interval (0, 1]. The inner product on this set is, � 1 �f, g� = f(t)g(t)dt. 0 Define f0(t) = 1. For e ach integer n ≥ 1, define fn(t) to be the piecewise continuous function whose 12 3value on (0,2n ] is −1, whose value on ( 21 n ,2n ] is +1, whose value on ( 2 ] is −1, whose value on 2n ,2n 3 (2n , 4 ] is +1, etc. In other words, 2n ⎧⎨ ⎩ 2k−2 < t ≤ 2k−2 for k = 1, . . . , 2n−1 ,−1, 2n 2n +1, 2n 2k−1 < t ≤ 2k for k = 1, . . . , 2n−1 .2n fn(t) = (a) Compute the integrals �fm, fn� and use this to prove that (f0, f1, . . . ) is an orthonormal sequence. a+1(Hint: If n > m, consider the integral of fn over one of the subintervals ( a 2m , 2m ]. What fraction of the time is fn positive and what fraction of the time is it negative?) (b) Compute the generalized Fourier coefficient, � 1 �t, fn(t)� = tfn(t)dt. 0 1 Prove it equals 2n+1 . This gives the generalized Fourier series, ∞t = 2n+1 n=0 (c) Rewrite the series above as, 1 fn(t). ∞.2n 2 n=1 1 1 + fn(t)t = What is the relationship of this equation to the binary expansion of the real number t?
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