Differential Equations Lecture Work Solutions 242

Differential Equations Lecture Work Solutions 242 - x ( t )...

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7. Solve the “signaling” problem v t + cv x =0 v (0 ,t )= G ( t ) −∞ <t< in the region x> 0. The easiest way is to reverse the role of x and t . So the problem is now v x + cv t =0 v ( x, 0) = G ( x ) −∞ <x< or v t + 1 c v x =0 v ( x, 0) = G ( x ) −∞ <x< The ODEs are dx dt = 1 c , dv dt =0 and the initial condition for each x (0) = ξ, v ( ξ, 0) = G ( ξ ) Solve the characteristic equation
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Unformatted text preview: x ( t ) = 1 c t + The solution of the other ODE v ( x ( t ) , t ) = G ( ) Solve for and substitute in v to get v ( x, t ) = G x 1 c t Now change the variables back v ( x, t ) = G t 1 c x . 242...
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