The charge on a capacitor q(t) in a circuit with a resistor, a capacitor and an inductor connected in series driven by a given time-dependent voltage v(t) is governed by the second-order ODE q"(t) + MU) + 5W) = "05)- (1) 1. (a) Use the unit step function, u(t), to express the time-dependent voltage 0, 0 2. in the form v (t) = f(t - a)u(t - a) -f(t-B)u(t -B), where the function f(t) and constants a, B are quantities that you need to determine. (b) Use your answer to Question 1(a), the linearity of property of Laplace transforms and the table of Laplace transforms on MyUni to find the Laplace transform of v(t). 2. Complete the square of the denominator and use the table of Laplace transforms on My Uni to find the inverse Laplace transform of the function 1 F(s) = 2+2s +5' 3. Use partial fraction decomposition, linearity and the table of Laplace transforms on My Uni to find the inverse Laplace transform of the function 1 H($) = $2 ($2 + 2s + 5) Hint: You are permitted to use computer algebra software (e.g. Wolfram Alpha) to find the partial fraction decomposition of H(s).4. Use the convolution theorem and your answer to Question 2 to verify the inverse Laplace transform of the function H (3) found in Question 3. i.e. Find the inverse Laplace transform of the function 1 1 3'3) = m = :2")- H int: You may use the following results [93 sin 2:1? d1: = %e_I ( sin(2.'c) + 2cos(2a:)) + C, '/:I:e_z sin 22: d2: = ire"(3 5.1:) sin(2;c) 2(53: + 2) cos(2:r,)) + C, where C is the constant of integration. 5. Use your answer to Question 3 to nd the inverse Laplace transform of the function e_"'"i K03) = 32(32 + 23 + 5), 7>0. 6. Use Laplace transforms and the results from Questions 1 5 to nd the response of (1) subject to the initial conditions q(0}=0 and (mi) =0. 7. Plot the response, q(t), found in Question 6, on the interval 0