test1 - ( 10 pts. ) u x + ( x + 1) u y = 0 [a] Sketch some...

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MIDTERM TEST I Course: APM346H Instructor: Chugunova Marina Date: 6 October 2008 This test consists of 4 problems. Please label clear all your sketches. Use the backs of pages if you need more room to write. No books, notes, or calculators allowed. The total number of points is 46. GOOD LUCK! Part 1. (Total 28 pts.) Quiz: Question 1. ( 10 pts. ) For each of the following equations state the order and whether it is nonlinear, linear inhomogeneous, or linear homogeneous. [ a ] u 3 t + x 2 u x + u = 0 [ b ] ( f ( x ) u tt ) x = 0 [ c ] 2 u x + u tx + 1 = 0 [ d ] xu x + f 00 ( t ) u t + t = 0 [ e ] xu x + tu xt + u = 0 Question 2. ( 8 pts. ) 2 u xx - 8 u xy + 8 u yy + u y = 0 [a] What is the type of the equation ? [b] Find a change of dependent variables which transforms the equation to a canonical form. [c] Justify your choice in [b]. Question 3.
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Unformatted text preview: ( 10 pts. ) u x + ( x + 1) u y = 0 [a] Sketch some characteristics. [b] Find the general solution. [c] Find the domain where exists the solution for the initial value problem: u ( x,x ) = sin( x ) , x ≥ 0. [d] What is special about the point x = 0 , y = 0 [e] Is the solution of this initial problem unique in the domain [c]? Justify your answer. Part 2. (Total 18 pts.) Problem: u x-2 xu y = 2 x y u, x > , y > [a] Sketch some characteristics. [b] Find the general solution. [c] Find the solution of the initial value problem: u (0 ,y ) = y 3 , 1 ≤ y ≤ 2. [d] Find and sketch the domain where the solution [c] exists and unique....
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This note was uploaded on 04/20/2011 for the course ENGINEERIN APM384 taught by Professor Chungonova during the Fall '11 term at University of Toronto.

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