Consider the boundary value problem involving Laplace’s equa…
Consider the boundary value problem involving Laplace’s equation in a semi-infinite slab { u x x + u y y = 0 , 0 0 , u ( 2 , y ) = 0 , y > 0 , u y ( x , 0 ) = 0 , 0 0 {“version”:”1.1″,”math”:”u(x,y)=A\cos(\omega x)e^{\omega y} + B\sin(\omega x)e^{\omega y} + C\cos(\omega x)e^{-\omega y} +D\sin(\omega x)e^{-\omega y},\ \omega >0″}where A, B, C, D{“version”:”1.1″,”math”:”A, B, C, D”} and ω{“version”:”1.1″,”math”:”ω”} are constants to be determined. Part (a): Find ALL values of ω{“version”:”1.1″,”math”:”ω”} that produce nonzero solution to the PDE and satisfies ALL the homogeneous BC. Also write their corresponding “eigenfunctions”. Part (b): Write a linear superposition of only the functions in the second answer box in part (a) above to use in part (c). Part (c): Apply the remaining nonhomogeneous BC to the answer you wrote in part (b) to find the function u(x,y){“version”:”1.1″,”math”:”u(x,y)”} that solves the full BVP. You must use the answer you wrote in part (b) above to get any credit here.
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