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Differential Equations Lecture Work Solutions 277

# Differential Equations Lecture Work Solutions 277 - = 1 2...

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2. u ( x, t )= F ( x 2 t )+ G ( x +2 t ) u ( x, 0) = cos x u t ( x, t )=0 u ( x, 0) = F ( x )+ G ( x )=co s x (*) u t ( x, t )= 2 F 0 ( x 2 t )+2 G 0 ( x +2 t ) u t ( x, 0) = 2 F 0 ( x )+2 G 0 ( x )=0 Integrate ⇒− F ( x )+ G ( x )=con s tan t= k (#) solve the 2 equations (*) and (#) 2 G ( x )=co s x + k G ( x )= 1 2 cos x + 1 2 k 2 F ( x )=co s x k F ( x )= 1 2 cos x 1 2 k We need F ( x 2 t ) F ( x 2 t )= 1 2 cos ( x 2 t ) 1 2
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Unformatted text preview: ) = 1 2 cos ( x + 2 t ) + 1 2 k ⇒ u ( x, t ) = 1 2 cos ( x − 2 t ) + 1 2 cos ( x + 2 t ) − 1 2 k + 1 2 k u ( x, t ) = 1 2 { cos ( x − 2 t ) + cos ( x + 2 t ) } 277...
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