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421s08x2h

421s08x2h - Math 421:01&02 Prof Bumby Exam 2 Name All class...

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Math 421:01&02 Prof. Bumby Exam 2 April 09, 2008 Name All class exams will be graded on the basis of 100 points. On this exam, those points are distributed over 6 problems (on 5 pages). No books or papers other than the official formula sheet may be used, but the formulas on that sheet may be used to find any quantity needed for your work. Calculators or other electronic devices are not allowed. You may leave when you are finished. 1. (15 pts.) Follow the steps below to use the method of separation of variables to find solutions u.x; y/ of x 2 u xx C 3xu x D u y : a . Assuming u.x; y/ D X.x/Y.y/ , find differential equations for X.x/ and Y.y/ b . Determine general solutions to these equations. X(x)= Y(y)= c . Find any solutions of the equations in (a) not covered by the formulas in (b). Also replace any expression containing complex numbers with suitable real functions.

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Math 421:01&02, April 09, 2008, p. 2 2. (20 pts.) Follow the steps outlined below to find the Fourier series of the function on the interval OE 3; 3Ł defined by f .x/ D 8 < : x C 3 for 3 < x < 2 1 for 2 < x < 2 3 x for 2 < x < 3 and shown in the graph a . Write an expression for the series using indeterminates a n or b n (as appropriate) for the coefficients, but identifying the functions appearing in the series. You may omit terms that you know will have coefficient zero. b . Calculate the coefficients (answers may be left in terms of trigonometric functions of explicit multiples of n )
Math 421:01&02, April 09, 2008, p. 3 3. (15 pts.) Find the first three terms c 0 P 0 .x/ C c 1 P 1 .x/ C c 2 P 2 .x/ of the Fourier-Legendre series of f .x/ D e 2x on OE 1; 1Ł . 4. (10 pts.) Consider the inner product h f; g i D Z 1 0 e x f .x/g.x/ dx and let f .x/ D x 4 C 4x 2 3 . Compute the following: h 1; f .x/ i D h x; f .x/ i D

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Math 421:01&02, April 09, 2008, p. 4 5. (20 pts.) Consider the heat equation for temperature u.x; t/ : 7u xx D u t with the constant temperature boundary condition, u.0; t/ D u.2; t/ D 0 , and initial condition u.x; 0/ D x for 0 < x < 2 . a . Use separation of variables to find functions of the form u.x; t/ D X.x/T .t/ that satisfy the differential equation and boundary conditions, and to write an expression for the series using indeterminates a n or b n (as appropriate) for the coefficients, but identifying the functions appearing in the series for u.x; t/ . b . Use the result of (a) to find the solution of the equation satisfying both the boundary conditions and the initial condition.
Math 421:01&02, April 09, 2008, p. 5 6. (20 pts.) Consider the wave equation for displacement u.x; t/ : 4u xx D u tt with the fixed end boundary conditions u.0; t/ D u.5; t/ D 0 and the initial conditions u.x; 0/ D x.5 x/ and u t .x; 0/ D 0 . a . Use separation of variables to find functions of the form u.x; t/ D X.x/T .t/ that satisfy the differential equation and boundary conditions, and to write an expression for the series using indeterminates a n or b n (as appropriate) for the coefficients, but identifying the functions appearing in the series for u.x; t/ .

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421s08x2h - Math 421:01&02 Prof Bumby Exam 2 Name All class...

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