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# L25 - The Wave Equation t For u(t x = T(t)X(x the...

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The Wave Equation t u ( t, L ) = 0 u ( t, 0) = 0 u tt = α 2 u xx 0 L x u (0 , x ) = f ( x ) u t (0 , x ) = g ( x ) For u ( t, x ) = T ( t ) X ( x ) the differential equation imp T ”( t ) X ( x ) = α 2 T ( t ) X ”( x ) = T 2 T ( t ) = X ”( x ) /X ( x ) = The boundary conditions im X (0) = X ( L ) = 0 To start with, find all funct X ( x ) and constants σ with X ”( x ) = σX ( x ) X (0) = X ( L ) = 0

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The situation is very similar to the first heat conduction problem we studied: We have a differential equation u tt = α 2 u xx , ( E ) two homogeneous boundary conditions u ( t, 0) = u ( t, L ) = 0 , ( BC ) and two initial conditions: u (0 , x ) = f ( x ) , ( IP ) and u t (0 , x ) = g ( x ) . ( IV ) As in the heat conduction case we go first after the three homogeneous Green con- ditions ( E ) , ( BC ) . 2
Again we try first for solutions of them that have their variables separated: u ( t, x ) = T ( t ) X ( x ) . ( SV ) Any linear combination of those will still satisfy ( E ) and ( BC ) , and among the latter we’ll then identify the one that satisfies ( IP ) and ( IV ) . ( SV ) turns ( E ) into T 00 ( t ) X ( x ) = α 2 T ( t ) X 00 ( x ) = T 00 ( t ) α 2 T ( t ) = X 00 ( x ) X ( x ) .

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L25 - The Wave Equation t For u(t x = T(t)X(x the...

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