s1 - Problem Set 1 1 a Let φ x y be a differentiable...

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Unformatted text preview: Problem Set 1 1. a. Let φ ( x, y ) be a differentiable function. Let a ( x, y ) = φ y and b ( x, y ) = φ x . Solve the PDE au x- bu y = 0 . Answer Let r ( t ) be a parametrization of a characteristic curve, so that dr/dt = φ y i- φ x j. As dr/dt · ∇ φ = 0, dr/dt is perpendicular to the gradient of φ . In 2d, this means that r is on a level curve of φ . Therefore, characteristic curves satisfy equations of the form φ ( x, y ) = constant . Using this constant to label the curves, one can write the general solution of the PDE as u = f ( φ ( x, y )) . where f is an arbitrary function. b. If φ = e xy , sketch some of the characteristic curves. Answer The curves are of the form xy = constant. Note that u is a single constant only on a connected piece of level curve. 2. (Problem 7 of Section 1.2 of the text book) Solve the PDE au x + bu y + cu = 0 . Answer The first step is to convert the equation into the form u ξ + cu = 0. Conparing ∂ ξ = ∂x ∂ξ ∂ x + ∂y ∂ξ ∂ y...
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s1 - Problem Set 1 1 a Let φ x y be a differentiable...

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