HW9 - is linear or non linear If it is linear state...

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Math 401 (Sec. 502) Spring, 2009 Homework 9 (due April 2) Q1. Verify that the following are regular S-L problems and then compute the eigenvalues and eigenfunctions for each of them: (a) y ′′ + λy = 0 , 0 x π and y (0) = 0 , y ( π ) = 0. (b) y ′′ + λy = 0 , 0 x 1 and y (0) = 0 , y (1) = 0. (c) y ′′ y + λy = 0 , 0 x 1 and y (0) = 0 , y (1) = 0. (d) y ′′ + λy = 0 , 0 x 1 and y (0) = 0 , y (1) + y (1) = 0. Q2. Consider the functions (i) y ( x ) 1 , 0 x 1, (ii) y ( x ) = 2 x 1 , 0 x 1. Find the generalized Fourier series expansions of these functions for the eigen- functions in Q1 (b). Q3. For each of the following, find the order and state whether the PDE
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Unformatted text preview: is linear or non linear. If it is linear, state wheather it is homogeneous or nonhomogeneous. (a) u xx + xu y = y (b) uu x − 2 xyu y = 0 (c) u 3 x + uu y = 1. (d) u xxxx + 4 u xxyy + u yyyy = 0 (e) u xx + 2 u xy + u yy = cos x (f) u xxx + u xyy + log u = 0 Q4. Show that u ( x, y ) = f ( x ) g ( y ) where f and g are arbitrary twice diﬀer-entiable functions satisﬁes uu xy − u x u y = 0 ....
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