221 Math Test 3
Spring 2009
Name: Score: Instructions: You must show supporting work to receive full and partial credits. No text book, personal notes, formula sheets allowed. One formula sheet is provided on next page. 1(20pts) Find the general solution of the equations x = -7x + 15y y = -6x + 11y (Hint: The characteristic equation is 2 - 4 + 13 = 0.) 2(20pts) You are given the fact that = 2 is the only eigenvalue and v = 2 is the only linearly independent 1 eigenvector of the system of equation below. Find the a generalized eigensolution of the system, and then find the general solution of the system. x = 4x - 4y y =x
3(20pts) Use definition only to find the Laplace transform of the function f (t) = (You can use the formula eat sin(bt)dt = sin(bt), 0, t<1 t1
b at a2 +b2 e
a at a2 +b2 e
sin(bt) -
cos(bt) + C.)
4(20pts) Find the Laplace transform or inverse transform: (a) Find L{t2 sin t}(s).
7s+10 (b) If { L-1 s3 +4s2 +5s }(t).
5(20pts) Use the Laplace method to solve the initial value problem x + 2x = f (t), x(0) = 1 with f (t) = 0, 2t,
1 t< 2 t 1. 2
1 (Hint: The solution is x(t) = e-2t + u(t - 2 )(t - 1 ). ) 2
Bonus 2pts: The University of Nebraska system was founded in the year of The End
Math 221 Test 3, Spring 2008
Page 2
Table of Laplace Transforms and Inverse Transforms f (t) = L-1 {F (s)}(t) F (s) = L{f (t)}(s)
tn eat eat sin bt eat cos bt eat f (t) u(t - a)f (t) u(t - a) f (t)
tt-a
n! , (s - a)n+1 b , (s - a)2 + b2 s-a , (s - a)2 ...