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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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- Winter '20