Explanation Given: The given boundary conditions are u ( x , 0 ) = 0 , u ( x , π ) = 0 and u ( 0 , y ) = 50 . Calculation: The partial differential equation is, ∂ 2 u ∂ x 2 = ∂ 2 u ∂ y 2 . (1) Consider the equation, u ( x , y ) = X ( x ) Y ( y ) . (2) Separate the variables of equation (2), − X ″ X = − λ and Y ″ Y = − λ The equation of the variable X is, X ″ − λ X = 0 . (3) The equation of the variable Y is, Y ″ + λ Y = 0 . (4) Substitute λ = α 2 in equation (3), X ″ − α 2 X = 0 The general solution of the above equation is, X ( x ) = c 1 e α x + c 2 e − α x . (5) At boundary condition x → ∞ , c 1 = 0 . For non-trivial solution c 2 ≠ 0 , α = n . Substitute c 1 = 0 and α = n in the equation (5), X ( x ) = c 2 e − n x . (6) Substitute λ = α 2 in equation (4), Y ″ + α 2 Y = 0 The general solution of the above equation is, Y ( y ) = c 3 cos α y + c 4 sin α y

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 12, Problem 9RE

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The steady state temperature u(x,y) in the semi-infinite plate.

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The steady state temperature u ( x , y ) in the semi-infinite plate.

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