Math 54 - Spring 1996 - Bergman - Make-up Final

# Math 54 - Spring 1996 - Bergman - Make-up Final - FRI 11:14...

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Unformatted text preview: 10/05/2001 FRI 11:14 FAX 6434330 MOFFITT LIBRARY 001 George M. Bergman Spring 1996, Math 54, Lee. 2 20 May, 1996 959 Evans Hall Make-up Final Exam 12:50-3:50PM 0 l O 1. (35 points) Consider the matrix A: —l —1 —l 8 3 4 (a) (12 points) Find the characteristic polynomial of A, the eigenvalues of A, and a basis for the eigenspace of each eigenvalue. (b) (12 points) Find the general solution to the differential equation x’ = Ax. (c) (7 points) Show that the system of two differential equations u” = —u'—u-—- v, v’ = 3u’+8u +412, can be reduced to a system of three ﬁrst—order differential equations, namely the system given in part (b) above; and use the result of (b) to obtain the general solution to the above equations in u and 1}. (Alternatively, you may use any method to solve the above system of two equations, if you show your work and get the right answer.) (d) (4 points) Obtain the particular solution to this system that satisﬁes u(0) = u’(0) = 0, 11(0) = 9. 2. (10 points) Let f and g be differentiable functions on an interval 1. (a) (3 points) Deﬁne the Wronskian, W(ﬁ g). (b) (7 points) Prove that if there is some xe I such that W( f, g)(x) i 0, then f and g are linearly independent functions; or equivalently, that if f and g are linearly dependent, then W( f, g)(x) = 0 for all x. 3. (15 points) Suppose p is a continuous function on the real line, and we wish to study functions u(x, y) satisfying the partial differential equation uyy(x, y) = POE) "Ax, y)- (a) (8 points) Obtain the conditions two functions a(x), b(y) must satisfy for the function u(x, y) = a(x) b(y) to be a solution to the above partial differential equation, in the form of a differential equation to be satisﬁed by each function, involving a constant in common. (This will be simpler than the considerations in Boyce and DiPrima for the heat and wave equations, because we are not imposing boundary conditions.) (b) (7 points) Find (using the result of (a) or any other method) some nonconstant solution to the partial differential equation uyy (x, y) = x ux(x, y). 10/05/2001 FRI 11:14 FAX 6434330 MOFFITT LIBRARY 002 1 1 4, (10 points) Let V be the subspace of C3 spanned by the vectors 1 and 0 0 —i Find an orthonormal basis of V with respect to the standard complex inner product on C3. 5. (12 points) Given that one solution to the differential equation xzylf _ = O is y1(x) : {1, ﬁnd a second solution 3720:), which is not a scalar multiple of y1 (x). 6.. (18 points) Let f be the function on the interval [0, 1] such that f(x) is 0 for OSxSl/z, and 1 for 1/2<xS 1. (a) (9 points)_ Find the coefﬁcients b” in the Fourier expansion for this f of the form 23:1 [7” sin nnx. (You may express these coefﬁcients using values of trigonometric functions, without evaluating these for each n.) (b) (5 points) Using the above result, ﬁnd the solution to the partial differential equation 2 ur(x,r) : at uxx(x,t) (0 < x <1, 0 <1) subject to the initial conditions u(x,0) = 0 for 0 < x < 1/2 and u(x,0) = 1 for 1/2 < x.< 1, and the boundary conditions u(0, t) = 0 : u(1, t) for t> 0. (c) (4 points) Sketch the function to which the series you found in part (a) converges on the interval [—2, +2], indicating with a heavy dot the value of this function at each point of discontinuity. ...
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## This note was uploaded on 04/02/2008 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at Berkeley.

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