# MA2051--C97 --- Solutions for TEST II

1. (a) The roots for the characteristic equation are -1 and 3/2, so two solutions are e-t and e(3/2) t. To verify independence, compute the Wronskian and show that it is not zero.
(b) Use the method of undetermined coefficients to find yp(t) = 1/2 e-3t.
(c) The general solution is

The initial data give c1 = -6/5 and c2 = +6/5, so y(t) = (6/5) e (3/2)t - (6/5) e- t + (1/2) e-3t.
(d) For large values of t, the solution approaches the exponential term (6/5)e(3/2)t -- everything else is transient. There is no limiting periodic solution. <\br><\br> 2. (a) The data give you m = 6 and k = (6)(9.8)/(0.02) = 2940, so the model reduces to

(b) The characteristic equation has roots (approximately) and so the general solution is . The initial data determine c1 = 0 and c2 = -0.03.
(c) The amplitude is 0.03. The period is and it takes one half of a period to travel from highest position to the lowest position.
(d) The velocity is and the mass passes through equilibrium at . The velocity is therefore (meters per second).

3. (a) The characteristic equation reduces to (r+4)(r-2) = 0 so the roots are r1 = -4 and r2 = +2. The first solution pair is (y1,z1) = (e-4t,3e-4t). The second solution pair is (y2,z2) = (e2t,e2t). (You can test linear independence with the Wronskian.) The general solution is (y,z) = c1(y1,z1) + c2(y2,z2).
(b) The initial data gives and so y= 50e2t, z= 50e2t. (This is a straight line and the solution moves away from the origin as t increases.)
(c) If you choose initial data so that the constants in the general solution become and c2 = 0, then you get convergence to the origin; the solution becomes (y,z) = c1(e-4t, 3 e-4t). (You must remove'' that nasty e+2t.) Any initial data of the form y(0) = (1/3) z(0) works. The solution curve in this case follows a straight line to the origin.

4. The critical damping coefficient is p = 4 and the solution in this case is

x(t) = xi e-2t + ( vi + 2xi) t e-2t

Set this equal to zero and solve for exactly one value:

If xi > 0, this value of t is positive if and only if vi < -2xi. (You have to push the mass toward the origin to be sure it gets there.)

Arthur Heinricher <heinrich@wpi.edu>