quiz4wedsolns.pdf - APM346 QUIZ 4 WEDNESDAY TUTORIALS 15 minutes Closed notes no external resources individual work Name Tutorial Student Number Quiz

# quiz4wedsolns.pdf - APM346 QUIZ 4 WEDNESDAY TUTORIALS 15...

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APM346 QUIZ 4, WEDNESDAY TUTORIALS 15 minutes Closed notes, no external resources, individual work Name: Tutorial: Student Number: Quiz policies : Indicate the main steps of your solution. This will increase your chances of getting at least partial credit. Recall D’Alembert’s formula : if u solves the homogeneous wave equation ( 2 t - 2 x ) u = 0, u | t =0 = g , t u | t =0 = h , with t, x R , then u ( t, x ) = 1 2 [ g ( x - t ) + g ( x + t )] + 1 2 Z x + t x - t h ( τ ) 1
2 APM346 QUIZ 4, WEDNESDAY TUTORIALS 1. (10 points) Solve explicitly the inhomogeneous

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Unformatted text preview: wave equation ( ∂ 2 t-∂ 2 x ) u = cos( t ) , t, x ∈ R , u | t =0 = 0 , ∂ t u | t =0 = cos( x ) . Solution. By the Duhamel principle and integration by parts, u ( t, x ) = 1 2 Z x + t x-t cos( τ ) dτ + 1 2 Z t Z x + t-s x-t + s cos( s ) dyds = 1 2 [sin( x + t )-sin( x-t )] + 1 2 Z t 2( t-s ) cos( s ) ds = 1 2 [sin( x + t )-sin( x-t )] + Z t sin( s ) ds = 1 2 [sin( x + t )-sin( x-t )] + 1-cos( t )...
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