For each of the following equations, write down the order of the equation, determine whether each of them is linear or non-linear, and say whether they are homogeneous or inhomogeneous. (i) e AU - y2024 Uxxy = 0. (ii) (1+U2)U – U₂Uy = 0. Find the general solutions U(x, t) for the PDE Uxx - 5Uxt + 4Utt = 0.
For each of the following equations, write down the order of the equation, determine whether each of them is linear or non-linear, and say whether they are homogeneous or inhomogeneous. (i) e AU - y2024 Uxxy = 0. (ii) (1+U2)U – U₂Uy = 0. Find the general solutions U(x, t) for the PDE Uxx - 5Uxt + 4Utt = 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
a and b please
Transcribed Image Text:(a) For each of the following equations, write down the order of the equation,
determine whether each of them is linear or non-linear, and say whether they are
homogeneous or inhomogeneous.
(i) e AU - y2024 Urxy = 0.
(ii) (1+U2)U - U₂Uy = 0.
(b) Find the general solutions U(x, t) for the PDE
Uxx - 5Uxt + 4Utt = 0.
(c) Consider the following 2nd order PDE,
Uxx + 2Uxy + 4U = 0.
I
(i) Write down its principal part and then determine the type (elliptic, parabolic
or hyperbolic).
(ii) Change the principal part of the above equation to a canonical form (i.e.
without cross-derivatives).
(d) Find all possible values of a, b, c so that U(x, t) = ax² + bt + ct² solves the heat
equation
Ut - Uzx = 0, x > 0.
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