Math 238

Find the roots of the corresponding auxiliary equation for the second order homogeneous dif- ferential equation y′′ + 2y′ − 3y = 0. A. r1 = 1, r2 = −3 B. r1 =−1,r2 =−3 C. r1 = 1, r2 = 3 D. r1 = −1, r2 = 3

A. r1 = 1, r2 = −3

The convolution of the functions f(t) = 1 and g(t) = cost is A. sint B. 1 C. cost D. 1−cost

A. sint

The integrating factor for the differential A. x^−3 B. x^3 C. x D. 3/x

A. x^-3

A mass-spring system has the following parameters: the mass m = 2 kg, the springs stiffness k = 2 N/m and a damping constant b N-sec/m. The system is under-damped if the damping constant satisfies A. b > 5 B. b < 4 C. b = 4 D. b = 5

B. b<4

Which differential equation is NOT separable? A. dy dx=xln(y2x)+7x2 B. dy dx= sin(x+y) C. dy dx= xyex+y D. dy dx= cos(x)/y

B. dy dx= sin(x+y)

Using the convolution theorem, the inverse Laplace transform L −1{ A. t∗cost B. sint∗t C. sint∗cost D. t2 ∗ sin t

B. sint∗t

Which differential equation is linear? A. y dy + xy = 2 dx B. d2y+2ydy+y=x dx2 dx C. d2y+2x2dy+(sinx)y=0 dx2 dx D. dy+y=siny dx

C. d2y+2x2dy+(sinx)y=0

Which one is an explicit solution to the given equation tdy = 2y dt A. y=e^2t B. y = t^3 C. y = t^2 D. y = e^t − t

C. y = t^2

If the roots of the auxiliary equation of a second order homogeneous differential equation with constant coefficients are r1 = 2 + i and r2 = 2 − i, then the general solution is A. y(t)=2cost+2sint B. y(t)=et(c1cos2t+c2sin2t) C. y(t)=e2t(c1cost+c2sint) D. y(t)=e−2t(c1cost+c2sint)

C. y(t)=e2t(c1cost+c2sint)

Consider the linear differential equation with constant coefficients, ay′′ + by′ + cy = g(t). Sup- pose that the the solution of the corresponding homogeneous equation is yh(t) = c1 cos t+c2 sin t and suppose that g(t) = (2t + 8) sin t. Then the particular solution has the form A. yp(t)=(A1t+A0B)sint+(B1t+B0)cost B. yp(t)=A1t+A0 +B0tsint C. yp(t)=(A1t+A0)tsint+(B1t+B0)tcost D. yp(t)=(A1t+A0)tsint.

C. yp(t)=(A1t+A0)tsint+(B1t+B0)tcost

The Laplace Transform of f (t) = t sin 2t is A. 2s/(s^2+4) B. 2/(s2 (s2 +4)) C. 2s/(s2 +4)^2 D. 4s/(s2 +4)^2

D. 4s/(s2 +4)^2

If y(x) = 3e2x − 4e−3x is the general solution of a certain second order homogeneous initial value problem, then the corresponding auxiliary equation is A. r2 − r + 6 = 0 B. r2 − 2r + 3 = 0 C. r2 + 2r − 3 = 0 D. r2 + r − 6 = 0

D. r2 + r − 6 = 0

Which pair of functions are NOT a linearly independent set of solutions to a second order differential equation? A. et cost and et sint B. tet and et C. et and e−t D. sin (2t) and sin t cos t

D. sin (2t) and sin t cos t

The inverse Laplace Transform of F (s) = 3/(s2 + 9) is A. cos(3t) √ B. sin( 3t) √ C. cos( 3t) D. sin(3t)

D. sin(3t)

Given that dy = 2x+y, y(0) = 1, step size h = 0.1, which of the following is the approximation dx to the given initial value problem at the points x = 0.1? A. y(0.1) = 1.01 B. y(0.1) = 1.11 C. y(0.1) = 1.21 D. y(0.1) = 1.10

D. y(0.1) = 1.10

Consider the linear differential equation with constant coefficients: y′′ + y′ = (t + 1)et. Then the particular solution has the form A. yp(t) = A1t + A0et B. yp(t) = A1t + A0 + B0tet C. yp(t) = (A1t + A0)tet D. yp(t) = (A1t + A0)et

D. yp(t) = (A1t + A0)et

Let y(t) be a function of t. Then (D + 3t)D[y] is, (Note: The differentiation is with respect to t) A. y′′ + 3y′ B. y′′ +3y′ +3t C. y′′ +3ty′ +3y D. y′′ + 3ty′

D. y′′ + 3ty′

Giventhaty′′−3y′+2y=0;y(0)=0,y′(0)=−1,whichofthefollowingisL{y}? A. s/((s+1)^2 +1) B. −1/(s2 +3s+2) C. 1/(s2 +2s+2) D. −1/(s2 −3s+2)

D. −1/(s2 −3s+2)