Question

3. (a) Using Laplace transforms, solve the following ordinary differential equation frac{d^2x}{dt^2} - 3frac{dx}{dt} + 2x = 2e^{-4t} subject to x = 0 and frac{dx}{dt} = 1 at t = 0. (10 Marks) (b) Consider the following periodic function f(t) given in one period: f(t) = egin{cases} t+1, & -1 < t le 1 \ 0, & -3 < t le -1 end{cases} (i) Sketch the function f(t) over the interval from -7 to 5. (ii) Find the Fourier series representation of the function f(t). (iii) Using the result obtained in part (ii), write down the partial sums of the series up to and including the term sin npi t. [Note: int t cos at,dt = frac{cos at}{a^2} + frac{tsin at}{a}, quad int t sin at,dt = frac{sin at}{a^2} - frac{tcos at}{a}] (15 Marks)

          3.
(a) Using Laplace transforms, solve the following ordinary differential equation
frac{d^2x}{dt^2} - 3frac{dx}{dt} + 2x = 2e^{-4t}
subject to x = 0 and frac{dx}{dt} = 1 at t = 0.
(10 Marks)
(b) Consider the following periodic function f(t) given in one period:
f(t) = egin{cases} t+1, & -1 < t le 1 \ 0, & -3 < t le -1 end{cases}
(i) Sketch the function f(t) over the interval from -7 to 5.
(ii) Find the Fourier series representation of the function f(t).
(iii) Using the result obtained in part (ii), write down the partial sums of the
series up to and including the term sin npi t.
[Note: int t cos at,dt = frac{cos at}{a^2} + frac{tsin at}{a}, quad int t sin at,dt = frac{sin at}{a^2} - frac{tcos at}{a}]
(15 Marks)

$3. (a) Using Laplace transforms, solve the following ordinary differential equation fracd^2xdt^2 - 3fracdxdt + 2x = 2e^-4t subject to x = 0 and fracdxdt = 1 at t = 0. (10 Marks) (b) Consider the following periodic function f(t) given in one period: f(t) = egincases t+1, -1 < t le 1 0, -3 < t le -1 endcases (i) Sketch the function f(t) over the interval from -7 to 5. (ii) Find the Fourier series representation of the function f(t). (iii) Using the result obtained in part (ii), write down the partial sums of the series up to and including the term sin npi t. [Note: int t cos at,dt = fraccos ata^2 + fractsin ata, quad int t sin at,dt = fracsin ata^2 - fractcos ata] (15 Marks)$

Added by Joseph W.

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