MA330F2005test2soln - M A33O Fall 2 005 Test 3 2 N ovember...

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MA33O Fall 2005 Test 3 2 November 2005 Name:
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1. Consider the initial value problem y" + 5y' + 6y : 12, g(0) : 0, y'(0) : 0. (a) Find the Laplace transform of the left hand side (LHS) of the above equation. (b) Find the equilibrium solution? (c) Solve the IVP using Laplace Transforms.
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3 (a) When using Fourierr series, we assume that a function /(r) that is periodic with period p:2L can be represented as an infinite series of the form f(*):"0*f, 11.,t Cos (9 + n=l b" Z\ ) ) * s ) n ( r y ) (b) /(r), is given graphically over one period, 1-f,a], Uy': t./n A good estimate for as would be /1. (c) Because /(r) is ""@1odd (circle one) function, we expect that the coefficients for t{e sine '; / cosine (circle oi?ferms should be 0. (d) Find the formula for the non-zero Fourier coefficients ft!.' ( t- *') dr (Add the necessary math symbols to complete the above expression so that it shows the standard form for the fourier series.) , x -+" cc'-(ry) ' ? = / / x- +- , \ t l , L A : C r / . - r * ' , r z - i l - * ' X \ ( o S n \ , . / - - 0 q " = s;.,(ry t u 1 r [ \ / '"--i- i \ " J / "*- | \ i - / l I / l I { r,s- {u€'} \ t " ) i t ) t l "1 I " l - J O
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4 3. Consider the one-dimensional wave equation, 't'111 : C2'l'Lrr,
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