23, Math Practice set for the midterm exam
1. The general solution of the dierential equation y + 2y = 3t2 e2t is: (a) 3t2 e2t + Ce2t (b) t3 e2t + C (c) t3 e2t + Ce2t (d) t3 e2t + Ce2t 2. The solution to the initial value problem y + 2y + 2y = 5ex , y(0) = 1, y (0) = 2, is: (a) y(x) = ex cos x + ex sin x + ex (b) y(x) = ex cos x + ex sin x + 5ex (c) y(x) = ex sin x + ex (d) y(x) = ex (cos x + 3 sin x) 3. Which of the following forms a set of fundamental solutions for the equation y 4y + 4y = 0. (a) y1 (x) = e2x , y2 (x) = e2x (b) y1 (x) = e2x , y2 (x) = (x + 1)e2x (c) y1 (x) = e2x , y2 (x) = 2xe2x (d) y1 (x) = e2x , y2 (x) = (x + 1)e2x 4. Which of the following is the correct form of a particular solution for the equation y 5y + 6y = cos 3x + e2x ? (a) Ae2x + B cos 3x + C sin 3x (b) Ae2x + Bx cos 3x + Cx sin 3x (c) Axe2x + Bx cos 3x + Cx sin 3x (d) Axe2x + B cos 3x + C sin 3x 5. Given that y1 (t) = t is a solution of t2 y + ty y = 0, t > 0, nd its general solution. 1
6. A dierential equation has the form y + p(t)y + q(t)y = 0, where p and q are both continuous on the interval 1 < t < 1. Explain why it is not possible that the general solution to that equation on that interval is y = At2 + Bt. 7. Consider the dierential equation
L[y]
t2 y t(t + 2)y + (t + 2)y = 2t3
(t > 0).
The functions y1 (t) = t and y2 (t) = tet are both solutions to the corresponding homogeneous equation L[y] = 0. Use the method of variation of parameters to nd a particular solution to the original equation. 8. A boater and a motor together boat weigh 640 lb. The motor provides a constant driving force of 200 lb in the direction of motion, and the resistance of the water is equal numerically to twice the velocity (in feet per second). If the boat is initially at rest, nd the time when the velocity equals to 50 ft/sec. (g = 32ft/sec2 ) 9. A box weighing 6.4 pounds is suspended from a spring hanging inside a uid-lled cylider. (This means that mg is equal to 6.4 pounds, where m is the mass of the box and g is the ft lb acceleration of gravity, 32 2 .) The spring constant is k = .6 . The damping constant, for sec ft lb sec . The box is hanging at rest, in the damping induced by the liquid in the cylinder, is .4 ft equilibrium. Then (say at time t = 0) the box is given a sharp tap that sends it downward at a speed of 2 feet per second. The box is observed to bob up and down with smaller and smaller bobs, approaching its rest position. (a) Find th...