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.
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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 - &lt; x &lt; m . b V $ &amp; 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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exam1su09 - respectively, find the solution of (*) which...

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