1 UNIVERSITI TEKNIKAL MALAYSIA MELAKA SESI PENGAJIAN 2-2011/12 DIFFERENTIAL EQUATIONS BENH 1253 ASSIGNMENT Answer all questions. 1. A Matter of Control For the equations in Problem (a) and (b), use each of the three controls: (i) u = 2(1 – x) (ii) u = -3 (iii) u = 2(1 – x) – 3 Determine and compare the response of the uncontrolled equation and the response of each controlled equation. For problem (a), x(0) = 1; for problem (b) x(0) = 1 and . (a) 2 + 3x = u (b) + + x = u [hint: use Maple or Matlab to sketch the graph] 2. Find the Laplace transform of the Problem 49 – 54 on text book page 290. 3. In an RL circuit as below the differential equation formed using Kirchhoff’s law, is defined by V dt di Ri ? L ? Given that R = 10 O, L = 3 H and V = 50 volts, and i(0) = 0. (a) Solve this differential equation by using separation of variables. (b) Sketch the graph of function, i ? f (t) . 4. In the RC circuit shown below, the switch is closed on position 1 at t = 0 and after 1 t is moved to position 2. (a) Find the complete current transient by using the integrating factor. (b) Sketch the graph of function for i ? f (t), m ?1, 2. m Hint: ? ???? ? ???? ? ? ? the time constant in the case of an RC circuit is = RC , and 1 Use the basic formula : ? dt dq q V i dt C dq R
2 5. Show that the given function are orthogonal on the indicated interval (a) f1(x) = x3, f2(x) = x2 + 1; [-1,1] (b) f1(x) = ex, f2(x) = sin(x); [p/4, 5p/4] 6. Find the Fourier series of f(x) = 7. Use separation of variables to find, if possible the solution for the given partial differential equation (a) , k > 0 (b) ux = uy + u 8. A string is stretched and secured on the x-axis at x = 0 and x = p for t > 0. If the transverse vibrations take place in a medium that imparts a resistance proportional to the instantaneous velocity, then the wave equation takes on the form Find the displacement u(x,t) if the string starts from rest from the initial displacement f(x)