Differential Equations Lecture Work Solutions 98

Differential Equations Lecture Work Solutions 98 - 1. utt =...

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1. u tt = c 2 ( u xx + u yy ) ¨ TXY = c 2 T ( X 0 Y + XY 0 ) ¨ T c 2 T = X 0 X + Y 0 Y = λ ¨ T + λc 2 T =0 X 0 X = Y 0 Y λ = µ X 0 + µX =0 Y 0 +( λ µ ) Y =0 a. X (0) = X 0 ( L )=0 Y (0) = Y ( H )=0 as in previous section u ( x, y, t )= X n =1 X m =1 ± a nm cos c ² λ nm t + b nm sin c ² λ nm t ³ sin n 1 2 π L x sin H y Initial Conditions f ( x, y )= u ( x, y, 0) = X n =1 X m =1 a nm sin n 1 2 π L x sin H y yields a nm a nm = R H 0 R L 0 f ( x, y )s in n 1 2 π L x sin H ydxdy R H 0 R L 0 sin 2 n 1 2 π L x sin 2 H ydxdy g ( x, y )= u t ( x, y, 0) = X n =1
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