Answer Question 1(A)

1. For the following questions, let A be the last non-zero digit of your student ID number and B be the second to last non-zero digit in your student ID number (for example, if your ID number is 001234567 then A = 7 and B = 6). PLEASE CLEARLY WRITE OUT THE VALUE OF YOUR A AND B ON THE FIRST PAGE OF YOUR SUBMISSION. (a) Find the general solution of the following linear fourth order homogeneous differential equation: di4 + (A2 - 2B2 ) d-4 + "diz + (2A2B) 4 + # + (B4 + A? B? )y = 0. HINT: Be careful when determining the auxiliary equation, and use the MATLAB/Octave roots () function to find the roots of the auxiliary polynomial. (8 marks) (b) Solve the following initial value problem using the Method of Undetermined Coefficients: day dt2 - 2A- dt Ady+ 5Amy = 30A sin(At), y(0) = 3, y'(0) = 0. (10 marks) (c) Use Laplace Transforms to solve the following Initial Value Problem: dy da- 2 dx -+ 2y = 2e-, y(0) = 2, y'(0) = -1. (12 marks) [30 MARKS TOTAL]