hw1Jan2007

hw1Jan2007 - (a) v ( x ) = 1 /x 2 (b) v ( x, t ) = t ( t +...

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CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 1 Distributed Wednesday January 10. Due Wednesday January 17, 2007 in class. 1. [30 points] Determine whether each of the following differential equations is an ODE or a PDE, determine its order, and specify whether it is linear or nonlinear. For those that are linear, specify whether they are homogeneous or inhomogeneous. (a) dv dx + 2 x v = 0 (b) ∂v ∂t - 3 ∂v ∂x = x - t (c) ∂u ∂t - ∂x ± 2 u ∂u ∂x ² = 0 (d) ∂u ∂t + u ∂u ∂x + 3 u ∂x 3 =0 (e) d 2 y dx 2 - μ (1 - y 2 ) dy dx + y =0 (f) d 2 dx 2 ± ρ ( x ) d 2 u dx 2 ² = sin( x ) 2. [30 points] Determine whether each of the followingfunctions is a solution of the corresponding differential equation in parts (a), (b), and (c) of question 1, respectively.
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Unformatted text preview: (a) v ( x ) = 1 /x 2 (b) v ( x, t ) = t ( t + x ) (c) u ( x, t ) = xe t 3. [20 points] Is there any constant f such that u ( t ) = e t is a solution of the ODE d 2 u dt 2 + 4 du dt-3 u = f ? If so, specify f . Otherwise, explain why no such f exists. 4. [20 points] Suppose that you have a solution u of the equation a ( t ) d 2 u dt 2 + b ( t ) du dt + c ( t ) u ( t ) = f ( t ) (1) and that v is a nonzero solution of the homogeneous equation a ( t ) d 2 u dt 2 + b ( t ) du dt + c ( t ) u ( t ) = 0 . Explain how to produce innitely many dierent solutions of (1)....
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