Differential Equations Lecture Work Solutions 180

# Differential Equations Lecture Work Solutions 180 - (a ∂...

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6.3 Canonical Forms Problems 1. Find the characteristic equation, characteristic curves and obtain a canonical form for each a. xu xx + u yy = x 2 b. u xx + u xy xu yy =0 ( x 0 , all y ) c. x 2 u xx +2 xyu xy + y 2 u yy + xyu x + y 2 u y =0 d. u xx + xu yy =0 e. u xx + y 2 u yy = y f. sin 2 xu xx +sin2 xu xy +cos 2 xu yy = x 2. Use Maple to plot the families of characteristic curves for each of the above. 3. Classify the following PDEs: (a) 2 u ∂t 2 + 2 u ∂x 2 + ∂u ∂x = e kt (b) 2 u ∂x 2 2 u ∂x∂y + ∂u ∂y =4 4. Find the characteristics of each of the following PDEs: (a) 2 u ∂x 2 +3 2 u ∂x∂y +2 2 u ∂y 2 =0 (b) 2 u ∂x 2 2 2 u ∂x∂y + 2 u ∂y 2 =0 5. Obtain the canonical form for the following elliptic PDEs:
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Unformatted text preview: (a) ∂ 2 u ∂x 2 + ∂ 2 u ∂x∂y + ∂ 2 u ∂y 2 = 0 (b) ∂ 2 u ∂x 2 − 2 ∂ 2 u ∂x∂y + 5 ∂ 2 u ∂y 2 + ∂u ∂y = 0 6. Transform the following parabolic PDEs to canonical form: (a) ∂ 2 u ∂x 2 − 6 ∂ 2 u ∂x∂y + 9 ∂ 2 u ∂y 2 + ∂u ∂x − e xy = 1 (b) ∂ 2 u ∂x 2 + 2 ∂ 2 u ∂x∂y + ∂ 2 u ∂y 2 + 7 ∂u ∂x − 8 ∂u ∂y = 0 180...
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