# HW12 - Math 401(Sec 502 Homework 12(due April 30 Spring...

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Math 401 (Sec. 502) Spring, 2009 Homework 12 (due April 30) Q1. Find the Green’s functions and use it to compute the solutions to the boundary value problems ( i ) u ′′ ( x ) = 1 , u (0) = 1 , u (1) = 1 (0 < x < 1) ( ii ) u ′′ ( x ) = x, u (0) = 2 , u (1) = 1 (0 < x < 1) ( iii ) u ′′ ( x ) = x, u (0) = 2 , u (1) = 1 (0 < x < 1) . Q2. (a) Find the Green’s function to the BVP u ′′ ( x ) = 1 k q ( x ) , u (0) = a, u (1) = b, (0 < x < 1) . (b) Prove that the solution to the above BVP is given by u ( x ) = 1 0 G ( x, ξ ) q ( x ) + bx + a (1 x ) . Q3. Determine the region in which the given equation is hyperbolic, parabolic or elliptic: ( i ) u xx + ( x 1) u xy + u yy 2 x 2 u x + 3 xyu y + 2 u = sin x ( i ) ( x + 2) u xx + 2( x + y ) u xy + 2( y 1) u yy 3 x 2 u x = x 3 y 3 ( iii ) x 2 u xx + 2 xyu xy + y 2 u yy = 0 , ( iv ) 4 u xx + 5 u xy + u yy + u x + u y = 2 . Q4. Transform the wave equation into the canonical form, and hence solve

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HW12 - Math 401(Sec 502 Homework 12(due April 30 Spring...

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