Differential Equations Lecture Work Solutions 278

Differential Equations Lecture Work Solutions 278 - 3. The...

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Unformatted text preview: 3. The wave equation utt − c2 uxx = 0 can be written as a system of two first order PDEs ∂v ∂v −c =0 ∂t ∂x and ∂u ∂u +c = v. ∂t ∂x Solving the first for v , by rewriting it as a system of ODEs dv =0 dt dx = −c dt The characteristic equation is solved x = −ct + x0 and then v (x, t) = v (x0 , 0) = V (x + ct) where V is the initial solution for v . Now use this solution in the second PDE rewritten as a system of ODEs du = V (x + ct) dt dx =c dt The characteristic equation is solved x = ct + x0 and then du = V (x + ct) = V (x0 + 2ct) dt Integrating u(x0 , t) = t 0 V (x0 + 2cτ )dτ + K (x0 ) Change variables z = x0 + 2cτ then dz = 2cdτ 278 ...
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