Winter 2006 - Reynold's Class - Quiz 1

# Winter 2006 - Reynold's Class - Quiz 1 - (b ∂y ∂t =...

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Math 20D, Lecture C, Winter 2006 20 January 2006 Quiz 1, version A Name: ID #: Section Time: Show all work clearly and in order, and circle your ﬁnal answers. You have 20 minutes to take this 25 point quiz. 1. [ 6 points ] An equilibrium of a diﬀerential equation is called stable if neighboring states are attracted to it, and is called unstable if the converse is true. Looking at the following direction ﬁeld, determine: (a) all equilibria of the diﬀerential equation (b) whether these equilibria are stable or unstable. Solution: y = 5 , unstable y = 3 , stable y = 2 , unstable 2. [ 9 points ] Classify the following diﬀerential equations according to each of the following three categories: (i) whether it is an ordinary or partial diﬀerential equation, (ii) the order of the diﬀerential equation, and (iii) whether it is a linear or nonlinear diﬀerential equation. (a) dy dt = cos( t ) y + e 2 t Solution: ﬁrst-order, linear, ODE

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Unformatted text preview: (b) ∂y ∂t = ∂y ∂x + y ∂ 2 y ∂t 2 Solution: second-order, nonlinear, PDE (c) ∂ 2 y ∂t 2 = 2 y e t + ∂y ∂x t 3 Solution: second-order, linear, PDE 3. [ 10 points ] Solve the following initial value problem: 3 dy dt =-3 t y + 3-1 t y (1) = 1 6 , t > Solution: Rewrite the equation as dy dt + 1 t y = 1-1 3 t Compute the integrating factor as [2 pts for setup, 2 pts for correct μ ( t ) ] μ ( t ) = exp ±Z t 1 1 s ds ² = exp h (ln s ) ³ ³ t 1 i = exp [ln t ] = t Use the integrating factor to compute the solution [2 pts for setup, 2 pts for correct y ( t ) up to a constant, 2 pts for the correct constant (to meet initial condition)] y ( t ) = 1 t ±Z t 1 s ´ 1-1 3 s µ ds + 1 6 ² = 1 t " ´ 1 2 s 2-1 3 s µ ³ ³ ³ ³ t 1 + 1 6 # = 1 t ± 1 2 t 2-1 3 t-1 2 + 1 3 + 1 6 ² = 1 2 t-1 3...
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## This note was uploaded on 06/12/2008 for the course MATH 20D taught by Professor Mohanty during the Spring '06 term at UCSD.

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Winter 2006 - Reynold's Class - Quiz 1 - (b ∂y ∂t =...

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