ExamIsummer08 mat325

# ExamIsummer08 mat325 - Mathematics 325 Exam I Summer 2008...

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Mathematics 325 Exam I Name: >. Grov~ Summer 2008 ex+ 9 )u @*,)GL+ 4, d,y - 1 .(25 pts.) Find the general solution of u, + Zu,, + (2x - y) u =% + 3xy - 2 y2 in the xy - plane. Bonus (10 pts.) Find the solution of this partial differential equation that satisfies auxiliary condition 5 u(x,O)=x+l-- for x > 0. 2x

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2.(25 pts.) Classify the following second-order partial differential equations as hyperbolic, parabolic, or elliptic. If possible, find the general solution of each in the xy -plane. 2 (a) u,+u,+3u,+u,=~ -+ Z~-~(IX. .)= -S~O 1-1 q+ 2 (b) um+u,-2u,+4u=0 4 B~~AC- ~+)-~cI)(I)= o [-I T,\,,

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3.(25 pts.) (a) Derive the general solution of u, - c2u, = 0 in the xt -plane. (b) Derive a formula for the solution of the partial differential equation part (a) wbich satisfies the initial conditions u(x, 0) = +(x) and u, (x, 0) = y/(x) for all real x. Here + ad y/ are two given "smooth" functions of a single variable. Bonus (10 pts.) Derive a general relation between + and wbich will produce a solution to the

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ExamIsummer08 mat325 - Mathematics 325 Exam I Summer 2008...

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