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課程名稱︰工程數學-微分方程(一)
課程性質︰電機工程學系大二共同必修
課程教師︰丁建均
開課學院：電機資訊學院
開課系所︰電機工程學系
考試日期（年月日）︰2019/11/6
考試時限（分鐘）：10:20-12:10 (110分鐘)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.Solve the following DE and give the corresponding largest interval for the
solution (if the explicit solution exists, please do not write as the
implicit solution, On the other hand, you may have to express the solution
in terms of an integral-defined function for some problem(s).)
(a) x (1+y^2)^(1/2) dx = y (1+x^2)^(1/2) dy (5%)
(b) dy/dx = exp(sqrt(x))/y , y(1) = 4 (5%)
(c) dr/dΘ + rsecΘ = cosΘ (5%)
(d) dy/dx+ exp(x)*y = 1 , y(0) = 1 (5%)
(e) dx/dy - xsin(y) = 2sin(y) , x(π/2) = 1 (5%)
2.Solve the exact equation:
dx/dy = -(4y^2 + 6xy)/(3y^2+2x) (5%)
3.Solve the following differential equations by using appropriate substitution
(a) dy/dx = (x + y + 1)^2 (5%)
(b) x(dy/dx) = y + sqrt(x^2-y^2) , x>0 (5%)
4.The mathematical model for the velocity v of a chain slipping off the edge
of horizontal platform is xv(dv/dx) + v^2 = gx , where x denote the
length of the chain overlies the table, v is its velocity, and g is the
gravity.
(a) Please rewrite the equation in the form of M(x,v)dx + N(x,v)dv
= 0 and then solve as a modified exact equation with an appropriate
integrating factor. (5%)
(b) Please rewrite the equation in the form of dv/dx + P(x)v =
f(x)v'' and then solve as a Bernoulli equation with an appropriate
substitution. (5%)
5.Suppose that in the temperature in a refrigerator is -7℃ and the temperatur
-e outside the refrigerator is 25℃. If we take a drinl out from the refrige
-rator, after 1 hour, the temperature of the drink is 17℃.
(a) Express how the temperature of the drink varies with time by a DE.(5%)
(b) Suppose that, after 1 hour, the temperature of the drink is 17℃.
How long will the temperature of the drink be 24℃? (5%)
6.Suppose that (x-2)y''(x) + xy'(x) + [(x+2)/(x+1)]y(x) = 0, y(Xo) = 0,
y'(Xo) = 1. In what conditions the DE may not have a unique solution(Show
the solutions in terms of Xo). (5%)
7.Solve the general solutions of the following DEs:
(a) (x+2)y''(x) - y'(x) = 0, y(x) = 1 is one of the solutions (5%)
(b) sin(x)y''(x) - cos(x)y'(x) = 0, y(x) = 1 is one of the solutions (5%)
8.For the following higher-order linear differential equations:
(a) Use the Auxiliary Equation Approach to determine the general
solution of y'''(x) - 6y''(x) + 11y'(x) - 6y(x) = 0. (7%)
(b) Use the Superposition Approach to determine the general solution of
y''(x) - 3y'(x) + 2y(x) = exp(3x). (7%)
(c) Use the Annihilator Approach to determine the general solution of
y''(x) - 3y'(x) + 2y(x) = exp(3x). (7%)
(d) Are these three general solutions you found for (a), (b), and (c)
the same? If yes, verify. If no, describe the reason. (4%)
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※ 編輯: chun10396974 (140.112.245.134 臺灣), 11/06/2019 15:14:25
※ 編輯: chun10396974 (140.112.245.134 臺灣), 11/07/2019 01:33:06