Ch13Sect4N16

# Ch13Sect4N16 - c F C c G D , which shows that u.x;t/ D 1 2...

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Mathematics 421 Solutions 13.4, Number 6 Spring 2008 The problem Solve u xx D u tt by d’Alembert’s method with the initial conditions u.x;0/ D 1 1 C x 2 ; u t .x;0/ D 0: and graph the result for t D 0;1;3 . Analysis We write u.x;t/ D F.x C t/ C G.x ± t/: Then u t .x;t/ D F 0 .x C t/ ± G 0 .x ± t/: Specializing to t D 0 , we have F.x/ C G.x/ D 1 1 C x 2 .0/ F 0 .x/ ± G 0 .x/ D 0 .1 ± / Differentiating equation .0/ gives F 0 .x/ C G 0 .x/ D ± 2x .1 C x 2 / 2 .1 C / Adding and subtracting equations .1 C / and .1 ± / gives F 0 .x/ D G 0 .x/ D ± x .1 C x 2 / 2 ; from which we get F.x/ D 1 2.1 C x 2 / C c F G.x/ D 1 2.1 C x 2 / C c G Checking these formulas in .0/ , we ﬁnd that

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Unformatted text preview: c F C c G D , which shows that u.x;t/ D 1 2 1 1 C .x C t/ 2 C 1 1 C .x t/ 2 : This is as simple an expression for the solution as you are likely to nd. Any further algebra would be a waste of time. Any plotting program should be able to produce a graph from this formula. Mathematics 421 Solutions 13.4, Number 6, p. 2 Graphs t D t D 1 t D 3...
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## This note was uploaded on 09/29/2009 for the course 650 421 taught by Professor Bumby during the Spring '08 term at Rutgers.

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Ch13Sect4N16 - c F C c G D , which shows that u.x;t/ D 1 2...

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