1. Using Laplace transforms, solve for x only in the following pair of simultaneous differential equations...

  

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1. Using Laplace transforms, solve for x only in the following pair of simultaneous differential equations dy 2 -y+x+ dt dx sin t = 0 dt dx dy 3. +x-y+ 2 dt dt given that x(0) = y(0) = 0. 2. Given f(x) = -x if -1 < x < 1 and f(x) = f(x + 2). %3D %3D (a) Find the trigonometric Fourier series for f(x). Expand the series up to the fourth harmonic. (b) Let x = 1/2 and deduce a series for a/4. Present your final answer using the Sigma notation. %3D 3. Given -A SI<0 f(1) = Cos t (a) Find the sum of the Fourier series at t = 0 (at the point of discontinuity). (b) Find the complex Fourier series for f(t). 1. Using Laplace transforms, solve for x only in the following pair of simultaneous differential equations dy 2 -y+x+ dt dx sin t = 0 dt dx dy 3. +x-y+ 2 dt dt given that x(0) = y(0) = 0. 2. Given f(x) = -x if -1 < x < 1 and f(x) = f(x + 2). %3D %3D (a) Find the trigonometric Fourier series for f(x). Expand the series up to the fourth harmonic. (b) Let x = 1/2 and deduce a series for a/4. Present your final answer using the Sigma notation. %3D 3. Given -A SI<0 f(1) = Cos t (a) Find the sum of the Fourier series at t = 0 (at the point of discontinuity). (b) Find the complex Fourier series for f(t).