Math 217 Fall 2000 Exam 3Notational Remark: In this exam, the symbol ∂∂x( )y x means dydx.A table of basic Laplace transform formulas may be found at the end of this exam.1. Suppose that = ( )f t 2 and = ( )g t t . Then what is the value ( )( )f * g 3 (where f * g is the convolution of the two given functions)? a) 1 b) 2 c) 3 d) 4 e) 6 f) 8 g) 9 h) 12 i) 32 j) 23 Solution: (g)> f := t -> 2; := f 2> g := t -> t;> f_convolve_g := t -> Int(f(tau)*g(t-tau), tau = 0 .. t);> f_convolve_g(3);d⌠⌡03 − 6 2 τ τ> value(%);92. The Laplace transform of Bessel's function is . What is the Laplace transform of ?a) −s + s21 b) s + s21 c) −s + s21 d) s + s21e) −1s + s21 f) 1s + s21 g) −s( ) + s2132 h) s( ) + s2132 i) −1s2 + s21 j) 1s2 + s21Solution: (h)> restart; with(inttrans);addtable fourier fouriercos fouriersin hankel hilbert invfourier invhilbert invlaplace, , , , , , , , ,[invmellin laplace mellin savetable, , , ]> laplace(t*f(t),t,s);−∂∂s( )laplace , ,( )f t t s> subs(laplace(f(t),t,s) = 1/sqrt(s^2+1), %);−∂∂s1 + s21> simplify(%);s( ) + s21( ) / 3 23. Evaluate d⌠⌡0t( )J0u ( )J0 − t u u by using the Laplace transform given in the preceding problem. a) ( )sin t b) ( )cos t c) ( )sin t ( )cos t d) ( )sin t2 e) ( )cos t2 f) ( )arcsin t g) ( )arccos t h) ( )arctan t i) ( )arcsin t2 j) ( )arccos t2Solution: (a)> restart: with(inttrans):> laplace(BesselJ(0,t),t,s)^2;1 + s21> invlaplace(%,s,t);( )sin t4. Let F be the Laplace transform of . What is ( )F 1 ? a) −16 b) 16 c) −13 d) 13 e) −23 f) 23 g) −38 h) 38 i) −12 j) 12 Solution: (j)> restart: with(inttrans):> laplace(sin(t),t,s);1 + s21> (-1)^2*diff(%,s$2); − 8s2( ) + s2132( ) + s212> normal(%);2 − 3 s21( ) + s213> subs(s=1,%);125. Suppose that = ( )g t {0 < t 1t2otherwise. If F is the Laplace transform of g, then what is e2( )F 2 ?a) 12 b) 32 c) 14 d) 34 e) 54 f) 74 g) 18 h) 38 i) 58 j) 78 Solution: (e)> series(t^2, t=1); + + 1 2 ( ) − t 1 ( ) − t 12> exp(-s)*(1/s+2/s^2+2/s^3);e( )−s + + 1s2s22s3> exp(2)*subs(s=2,%);54e2e( )-2> simplify(%);54 6. Let f be the periodic function with period 2 that is defined on the interval [0,2) by the formula = ( )f t {1 and ≤ 0 t < t 10 and ≤ 1 t < t 2. What is the Laplace transform of f ?a) + 1 e( )−s( ) − 1 e( )−2 ss b) − 1 e( )−s( ) − 1 e( )−2 ss c) − 1 e( )−s − 1 e( )−2 s d) + 1 e( )−s − 1 e( )−2 s e) e( )−s − 1 e( )−2 s f) s − 1 e( )−2 s g) s2 − 1 e( )−2 s h) − 1 s − 1 e( )−2 s i) + 1 s − 1 e( )−2 s j) 1( ) − 1 e( )−2 s( ) + s 1 Solution: (b)> 1/(1-exp(-2*s))*int(1*exp(-s*t),t=0..1);− − e( )−s1( ) − 1 e( )−2ss 7. Suppose that ( )x t is the solution of the initial value problem , , = + ∂∂2t2( )x t 4 ( )x t ( )δ1t = ( )x 0 0 = ( )( )D x 0 0 . What is the Laplace transform ( )X s of ( )x t ?a) ( )δ1s + s24 b) 2 ( )δ1s + s24 c) 4 ( )δ1s + s24 d) + 2 ( )δ1s + s24 e) 2 e( )−s + s24 f) 2 es + s24 g) + 2 e( )−s + s24 h) + 2 es + s24 i) e( )−s + s24 j) es + s24Solution: (i)> diff(x(t), t$2)+4*x(t) = Dirac(t-1); = + ∂∂2t2( )x t 4 ( )x t ( )Dirac − t 1> map(z->laplace(z,t,s),%); = − + s ( ) − s ( )laplace , ,( )x t t s ( )x 0 ( )( )D x 0 4 ( )laplace , ,( )x t t s e( )−s> subs({x(0)=0,D(x)(0)=0}, %); = + s2( )laplace , ,( )x t t s 4 ( )laplace , ,( )x t t s e( )−s> solve(%,laplace(x(t),t,s));e( )−s + s24 8. What is the determinant of the matrix 1 1 21 0 12 1 -1 ?a) 1 b) 2 c) 3 d) 4 e) 5 f) -1 g) -2 h) -3 i) -4 j) -5Solution: (d)> restart: with(linalg):Warning, the protected names norm and trace have been redefined and unprotected > det(matrix([[1, 1, 2], [1, 0, 1], [2, 1, -1]]));49. When the second order equation = + + x ( )'' t 2 x ( )' t 3 ( )x t 0 is converted to a first ordersystem = ( )x1t( )x2t' A( )x1t( )x2t with = x1x, what is A? a) 1 03 2 b) 1 0−3 −2 c) 1 1−2 −3 d) 2 33 2 e) 2 3−3 −2 f) 0 1−3 −2 g) 0 13 −2 h) 0 13 2 i) 1 13 2 j) 1 1−3 −2 Solution: (f) 10. What are the eigenvalues of −8 −105 7? a) {2,3} b) {2,-3} c) {-2,3} d) {-2,-3} e) {-1,3} f) {1,3} g) {1,-3} h) {-1,-3} i) {1,-1} j) {3,-3}Solution: (b)> eigenvals(matrix([[-8, -10], [5, 7]]));,2 -311. Find two distinct eigenvectors [1,a] and [1,b] of the matrix −5 4−6 5. What is the sum a + b ?a) 0 b) 1/2 c) 1 d) 3/2 e) 2 f) 5/2 g) 3 h) 7/2 i) 4 j) 9/2Solution: (f)> eigenvects(matrix([[-5, 4], [-6, 5]]));,[ ], ,-1 1 { }[ ],1 1, ,1 1 { },132> 1 + 3/2;52 12. If = u31 and = v52 are eigenvectors of = A−4 15−2 7, then what is the general solution of the system = x ' A x (where = xx1x2)?a) + c1u c2v b) + c1e( )−tu c2e( )2 tv c) + c1e( )2 tu c2e( )−tv d) + c1e( )−tu c2etv e) + c1etu c2e( )−tv f) + c1e( )2 tu c2etv g) + c1etu c2e( )2 tv h) ( ) +
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