Differential Equations Lecture Work Solutions 201

Differential Equations Lecture Work Solutions 201 - α x =...

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5. b. 2 u ∂x 2 2 2 u ∂x∂y +5 2 u ∂y 2 + ∂u ∂y =0 A =1 ,B = 2 ,C =5 ,D =0 ,E =1 ,F = G =0 . The discriminant ∆ = ( 2) 2 4 · 1 · 5=4 20 = 16 < 0 Therefore the problem is elliptic. The characteristic equation is dy dx = B ± 2 A = 2 ± 4 i 2 = 1+2 i The solutions are y = x ± 2 ix + C The transformation is ξ = y + x +2 ix η = y + x 2 ix In elliptic problems we use another transformation (to stay with real functions) α = y + x β =2 x Since these are linear functions, we only need the Frst partials
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Unformatted text preview: α x = 1 , α y = 1 β x = 2 , β y = 0 Use the formulae for A ∗ , B ∗ etc with α for ξ and β for η , we have A ∗ = 1 · 1 2 − 2 · 1 · 1 + 5 · 1 2 = 1 − 2 + 5 = 4 B ∗ = 0 as should be for elliptic C ∗ = A ∗ = 4 D ∗ = 1 · 1 = 1 E ∗ = 1 · 0 = 0 F ∗ = G ∗ = 0 Thus the canonical form is 201...
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