MidTerm1mathODE.pdf - (2015.10.21 1-3 2-1 2-2 2-4 2-5 2-6 2-8 Total score(105 points 1(20 points(1 Determine the order of the given dierential

MidTerm1mathODE.pdf - (2015.10.21 1-3 2-1 2-2 2-4 2-5 2-6...

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台師大數學系 微分方程第一次期中考 (2015.10.21) 考試範圍︰ 1-3; 2-1; 2-2; 2-4; 2-5; 2-6; 2-8 Total score (105 points) 1 (20 points) (1) Determine the order of the given differential equations; (2) state whether the equation is linear or nonlinear, and (3) state that the equation is ordinary or partial differential equation. (a) u t + uu x = 6 + u xxx , (b) dy dt + ty 3 = 0 , (c) u xxxx + u xyx + u yyy = 0 , (d) d 2 y dt 2 + sin ( t + y ) = cos t 2 (15 points) Solve the initial value problem (IVP) dy dx = 4 x + 2 2( y - 1) , y (0) = 3 2 , and determine the interval in which the solution exists. If its initial value is replaced by 1 2 , solve the IVP and its interval. - 2.0 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Page 1 Total pages: 2
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台師大數學系 微分方程第一次期中考 (2015.10.21) 3 (10 points) Consider the initial value problem dy dx - y = x + e x , y (0) = y 0 . Find the value of y 0 that separates solutions that grow positively as x → ∞ from those that grow negatively. How does the solution that corresponds to this critical value of y 0 behave as x → ∞ 4
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