Differential Equations Lecture Work Solutions 232

Differential Equations Lecture Work Solutions 232 - f ( x )...

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1. The PDE can be rewrriten as a system of two ODEs dx dt = 3 dw dt =0 The solution of the Frst gives the characteristic curve x +3 t = x 0 and the second gives w ( x ( t ) ,t )= w ( x (0) , 0) = sin x 0 =s in ( x +3 t ) w ( x, t )=s in ( x +3 t ) 2. a. The ODEs in this case are dx dt = c du dt = e 2 x Solve the characteristic equation x = ct + x 0 Now solve the second ODE. To do that we have to plug in for x du dt = e 2( x 0 + ct ) = e 2 x 0 e 2 ct u ( x, t )= e 2 x 0 1 2 c e 2
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Unformatted text preview: f ( x ) = u ( x , 0) = 1 2 c e 2 x + K Therefore K = f ( x ) 1 2 c e 2 x Plug this K in the solution u ( x, t ) = 1 2 c e 2 x + 2 ct + f ( x ) 1 2 c e 2 x Now substitute for x from the characteristic curve u ( x, t ) = 1 2 c e 2 x + f ( x ct ) 1 2 c e 2( x ct ) 232...
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