Explanation Given information: The u ( x , y ) = cos ( a x ) e b y is harmonic Concept used: The given polynomial u ( x , y ) is said to be harmonic iff ∇ 2 u = 0 ⇒ ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 Calculation: The polynomial, u ( x , y ) = cos ( a x ) e b y is harmonic It means ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 u x x + u y y = 0 Since u ( x , y ) = cos ( a x ) e b y Differentiating both sides with respect to x , we get, u x = ∂ ∂ x cos ( a x ) e b y ⇒ u x = e b y ∂ ∂ x cos ( a x ) = e b y ( − sin a x ) × a ⇒ u x = − a e b y sin a x Again differentiating both sides with respect to x , we get, u x x = − a e b y ∂ ∂ x sin a x ⇒ u x x = − a e b y cos a x × a ⇒ u x x = − a 2 e b y cos a x Now we will differentiate u ( x , y ) = cos (

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Author: Rogawski

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Chapter 14.3, Problem 78E

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constants a, b such that u(x,y)=cos(ax)eby is harmonic.

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Chapter 14.3, Problem 77E

Chapter 14.3, Problem 79E

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Tofind: constants a, b such that u ( x , y ) = cos ( a x ) e b y is harmonic.

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Tofind: constants a, b such that u(x,y)=cos(ax)eby is harmonic.

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Chapter 14 Solutions

Tofind:

constants a, b such that u(x,y)=cos(ax)eby is harmonic.