Instructions. Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without supporting work. A table of Laplace transforms and a convolution table have been appended to the exam. 6. Let A. 8 6 (a) [3 Points] Verify that the characteristic polynomial of A is c A (s) s +4. c A (s) det(si A) det (b) [ Points] Compute the matrix exponential e At. s 6 (s 6)(s+6)+4 s +4. 8 s+6 Since c A (s) s +4, (si A) s+6 s +4 8 s 6 [ s+6 s +4 8 s +4 s +4 s 6 s +4 ]. Then e At L { (si A) } [ cost+3sint sint ]. 4sint cost 3sint (c) [7 Points] Solve the initial value problem y Ay, y() 3 y e At y() [ cost+3sint sint ] 3 3cost+4sint. 4sint cost 3sint][ cost 6sint (d) [ Points] Solve the initial value problem y Ay + f(t), y() sint f(t)., where Math 7 Section November, 4
y(t) e At y()+e At f(t) [ cost+3sint sint ] 4sint cost 3sint [ cost+3sint + sint ] 4sint cost 3sint (cost) sint+3(sint) sint 4(sint) sint [ tsint+ 3(sinttcost) ] 4, (sinttcost) sint where the convolutions are computed from numbers 4 and 6 in the Table of Convolutions.. Consider the following first order linear system of differential equations y 7y +y y 4y +3y (a) [ Points] Write the system in matrix form y Ay. [ y y ] [ 7 4 3 ][ y y ]. (b) [ Points] Solve this system with the initial conditions y () c, y () c. 7 First compute e At, where A. We will use Fulmer s 4 3 method. The characteristic polynomial of A is s 7 c A (s) det(si A) det (s 7)(s 3)+4 s s+ (s ). 4 s 3 Thus, the only root is with multiplicity, so that B ca (s) {e t, te t }. Thus and differentiating gives e At M e t +M te t, Ae At M e t +M (e t +te t ). Evaluating at t gives the equations for M and M : I M A M +M. Math 7 Section November, 4
Thus, M I and M A I so that e At Ie t +(A I)te t e t + 4 e t +te t te t 4te t e t te t. te t Then the solution of the system is y (t) e At y () y (t) y () e t +te t te t c 4te t e t te t c c e t +(c +c )te t c e t +( 4c c )te t. 3. Let f(x) be the periodic function of period 4 that is defined on the interval (, ] by f(x) +x. (a) [ Points] Sketch the graph of f(x) on the interval [ 6, 6]. y 4 3 6 4 3 x 3 4 6 (b) [4 Points] Compute the Fourier series of f(x). The following integration formulas may be of use: xsinaxdx a sinax x a cosax xcosaxdx a cosax+ x a sinax. The period of f(x) is 4, so L. Then a f(x)dx (+x)dx ) (x+ x 4. Math 7 Section November, 4 3
For n, and a n b n f(x)cos n xdx (+x)cos n xdx cos n xdx+ f(x)sin n xdx (+x)sin n xdx sin n xdx+ + xsin n xdx [ 4 n sin n n xcos n ] x 4 n Thus, the Fourier series of f(x) is 4( )n+ cosn. n xcos n xdx xsin n xdx f(x) + 4 ( ) n+ sin n n x. n (c) [6 Points] Let g(x) denote the sum of the Fourier series found in part (b). Compute g(), g(), and g(). f is continuous at x so the Fourier series converges to f(x) at x. Thus, g() f(). For x, g() f(+ )+f( ) +4 f is continuous at x so g() f() f() 3.. 4. Consider the function f(x), < x <. Math 7 Section November, 4 4
(a) [4 Points] Let g(x) be the even periodic extension of f(x). Draw the graph of g(x) on the interval [, ]. (b) [4 Points] Let h(x) be the odd periodic extension of f(x). Draw the graph of h(x) on the interval [, ]. (c) [ Points] Compute the Fourier sine series of f(x). L so b n sinnxdx n cosnx ( cosn +) n n ( ( )n ) { if n is even n if n is odd. Thus, the Fourier Sine series of f(x) is f(x) n odd n sinnx. Math 7 Section November, 4
Laplace Transform Table f(t) F(s) L{f(t)}(s).. t n 3. e at 4. t n e at. cos bt 6. sin bt 7. e at cosbt 8. e at sinbt 9. h(t c) s n! s n+ s a n! (s a) n+ s s +b b s +b s a (s a) +b b (s a) +b e sc. δ c (t) δ(t c) e sc s Laplace Transform Principles Linearity Input Derivative Principles First Translation Principle Transform Derivative Principle L{af(t)+bg(t)} al{f}+bl{g} L{f (t)}(s) sl{f(t)} f() L{f (t)}(s) s L{f(t)} sf() f () L{e at f(t)} F(s a) L{ tf(t)}(s) d ds F(s) Second Translation Principle L{h(t c)f(t c)} e sc F(s), or L{g(t)h(t c)} e sc L{g(t+c)}. The Convolution Principle L{(f g)(t)}(s) F(s)G(s). Math 7 Section November, 4 6
Table of Convolutions f(t) g(t) (f g)(t). g(t) t g(τ)dτ. t m t n m!n! (m+n+)! tm+n+ 3. t sin at 4. t sinat. t cos at at sinat a a 3(cosat ( a t )) cosat 6. t cosat a 3(at sinat) 7. t e at e at (+at) a 8. t e at (a+at+ a t a )) 3(eat 9. e at e bt b a (ebt e at ) a b. e at e at te at. e at sinbt. e at cosbt 3. sin at sin bt 4. sin at sin at. sin at cos bt 6. sin at cos at 7. cos at cos bt 8. cos at cos at a bcosbt asinbt) a +b (beat acosbt+bsinbt) a +b (aeat b a(bsinat asinbt) a b a (sinat atcosat) b a(acosat acosbt) a b tsinat a b(asinat bsinbt) a b a (atcosat+sinat) Math 7 Section November, 4 7