In all the exercises, show your working and give your final answer correct to four decimal...

Transcribed Image Text:

In all the exercises, show your working and give your final answer correct to four decimal places if required. 1. Using Laplace transforms, solve for x only in the following pair of simultaneous differential equations [8] dy 22- y+x+ dt dx - sin t = 0 dt dy 3 +x-y+2 -e = 0 dx dt dt given that x(0) = y(0) = 0. 2. Given f(x) = -x if -1 <x< 1 and f(x) = f(x+ 2). (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 1/4. Present your final answer using the Sigma notation. [6] [4] 3. Given -ASI<0 f() = { cos! (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] [6] In all the exercises, show your working and give your final answer correct to four decimal places if required. 1. Using Laplace transforms, solve for x only in the following pair of simultaneous differential equations [8] dy 22- y+x+ dt dx - sin t = 0 dt dy 3 +x-y+2 -e = 0 dx dt dt given that x(0) = y(0) = 0. 2. Given f(x) = -x if -1 <x< 1 and f(x) = f(x+ 2). (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 1/4. Present your final answer using the Sigma notation. [6] [4] 3. Given -ASI<0 f() = { cos! (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] [6]