Assignment 4 Due: Fri. Mar. 10, 2000
1. Solve the general solution to the following second-order differential equations:
(i) 3x´´+ x´ - 10x = 0;
(ii) x´´ + 16x = sin(4t);
(iii) 2x´´ + 2x´ +5x = t2 + 2;
(iv) x´´ + 10 x´ + 25x = 2te-4t ;
(v) x´´ + 3x´ + 2x = 3e-2t
2. Find the explicit solution to the following initial value problems:
(i) x´´ + 5x´ + 6x = 0, x(0) = 2, x´(0) = -3;
(ii) x´´ + 4x = e-t sin2t, x(0) = 0, x´(0) = 1;
(iii) x´´ + 6x´ + 9x = 18, x(0) = 2, x´(0) = 4;
(iv) 4x´´ + 12x´ + 9x = e-3t/2, x(0) = -1, x´(0) = 1/2.
3. An Electrical RLC circuit has an inductance L = 2 henrys and a resistance R = 2 ohms and a capacitance of 1 farad. If the applied voltage E = t e-t volts find the current I (in amperes) in the circuit if I(0) = 0 and Q(0) = 2 coulombs. Find the current if the applied voltage is E = sin(t/2) e-t/2. Plot both these solutions using the MAPLE plot command given in Assignment 2.
4. Convert the following systems of first-order differential equations into a second-order differential equation in one of the dependent variables and thus find the general solution to the system:
|(i) x´ = 3x - 2y||(ii) x´ = 5x -2y + 3||(iii) x´ = -4x - 6y + 6e2t||(iv) x´ + y´ + 2x = sin t|
|y´ = 2x - 2y||y´ = 6x - 2y||y´ = x + y + 3e2t||2x´ - 2y´ - y = 0|
5. (a) Use MAPLE to plot the several orbits of the systems given in question 4 (i) and (ii)
(b) Convert the second-order differential equation in question 2 (i) into a system of first-order differential equations and use MAPLE to plot several orbits. Include the orbit corresponding to the initial conditions x(0) = 1, x´(0) = -2 and identify this orbit on the plot.
TERM TEST 2 - Monday, March 13