MATH 308 Summer 2008 Practice Test II Solutions, Exams of Differential Equations

Download MATH 308 Summer 2008 Practice Test II Solutions and more Exams Differential Equations in PDF only on Docsity! MATH 308 Summer 2008 Practice Test II 1. The Existence and Uniqueness Theorem guarantees that the solution to x3y′′ + x sin x y′ − 2 x − 5 y = 0, y(2) = 6, y ′(2) = 7 uniquely exists on (a) (−π, π) (b) (0, π) (c) (5, ∞) (d) (0, 5) 2. All of the following pairs of functions form a fundamental set of solutions to some second order differential equation on (−∞, ∞) EXCEPT (a) 1, e−t (b) cos t, sin(t + 2π) (c) e−2t cos 2t, e−2t sin 2t (d) e5t, e5t−1 3. Which of the following will be a particular solution to the equation 4y′′ + 4y′ + y = 24xe x 2 ? (a) x2(Ax + B)e x 2 (b) (Ax + B)e x 2 (c) x(Ax + B)e x 2 (d) (Ax + B) sin x 2 + (Cx + D) cos x 2 4. All the following differential operators are linear EXCEPT (a) L[y] = y′′ − 3y′ + y3 (b) L[y] = y′′ + y′ + 2y (c) L[y] = y′′ + sin xy′ + cos xy (d) L[y] = y′′ + xy′ + (x − 1)y 5. The motion of the mass-spring system with damping is governed by y′′ + 2y′ + y = 0, y(0) = 1, y′(0) = −3. This motion is (a) undamped (b) underdamped (c) critically damped (d) overdamped 6. e2+ 3π 4 i = (a) π (b) √ 2 2 (1 + i)e2 (c) √ 2 2 (1 − i)e2 (d) √ 2 2 (−1 + i)e2 7. A 2-kg mass is attached to a spring with stiffness k = 50 N/m. The mass is displaced 1/4 m to the left of the equilibrium point and given a velocity of 1 m/sec to the left. The damping force is negligible. The amplitude of this vibration is (a) √ 41 20 (b) 1 (c) √ 20 41 (d) 1 4 8. The FSS to the equation y′′ − 2y′ + 5y = 0 is (a) {cos x, sin x} (b) {ex cos 2x, ex sin 2x} (c) {ex, xex} (d) {ex, e−x}