exam1su09

exam1su09 - respectively find the solution of which...

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Mathematics 325 Exam I Name: %. &J 1 .(25 pts.) Find all seconddegree polynomial functions of two real variables, u(x,t)=m2 +bxt+ct2 +clk+et+ f where a, b, c, d, e, and f are real constants, which are solutions in the xt -plane of the one- dimensional diffusion equation 24,-Krr,=o. j rht) ZC ts, 4-0 her<. : a(?+zkt)+dx+f/ 7

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2.(25 pts.) Find the general solution of Jq-xU,, =o in the xy - plane. Sketch several characteristic curves of this partial differential equation.
3.(25 pts.) Consider the linearized gas dynamics equations where po is the density and co is the speed of sound in still air. Verify that if curl (v) = 0 when t = 0, then curl (v) = 0 at all later times.

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4.(25 pts.) Consider the partial differential equation (*I u, -324, -422" =o. (a) Classify (*) as elliptic, hyperbolic, or parabolic. (b) Find the general solution of (*) in the xt -plane. (c) If 4 and are C' and C' functions of a single real variable,

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Unformatted text preview: respectively, find the solution of (*) which satisfies the initial conditions u ( x , o ) = ~ ( x ) and u,(x,o)=y(x) forall - < x < m . b V \$ & I M lo p. 4-0 k w e . 1 Bonus.(25 pts.) A homogeneous solid material occupying D = {(x, y, z) E B3 : 4 i x2 + y2 + z2 i 100) is completely insulated and its initial temperature at position (x, y, z ) in D is 200/,/-. (a) Write (without proof or derivation) the partial differential equation and initialhoundary conditions that completely govern the temperature u (x, y, z,t ) at position (x, y, z) in D and time t 2 0. @) Use Gauss' divergence theorem to help show that the heat energy H (t) = IIIcpu (x, y, z, t)dv of D the material in D at time t is a constant function of time. Here c and p denote the (constant) specific heat and mass density, respectively, of the material in D. (c) Compute the (constant) steady-state temperature that the material in D reaches after a long time. 7....
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This note was uploaded on 01/11/2011 for the course MATH Math 325 taught by Professor Davidfrow during the Fall '10 term at Missouri S&T.

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exam1su09 - respectively find the solution of which...

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