課程名稱︰工程數學一
課程性質︰必修
課程教師︰李克強
開課學院：工學院
開課系所︰化學工程學系
考試日期（年月日）︰2012/11/09
考試時限（分鐘）：130
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(一) Solve the following first order ODE's:
(5%) (1) y' - y tan x = x
(5%) (2) 2xy y' - y^2 = x^2
(5%) (3) y" - y' - 2y = 0
(5%) (4) y''' - y = 0
(5%) (5) y" + y = sin x
(5%) (6) y" - y = x exp(x)
(5%) (7) x^2 y" + 2x y' - n(n+1) y = 0
(10%) (8) x^2 y" + x y' - y = x
(10%) (9) x^2 y" - 2x y' + 2y = x ln x
(5%) (10) y" = x (y')^3
(5%) (11) y" + exp(2y) (y')^3 = 0
(5%) (12) (1+x^2) dy + (1+y^2) dx = 0
(5%) (13) dy 1+y^2+3(x^2)y
-- = ------------
dx 1-2xy-x^3
(5%) (14) dy x+y-5
-- = -----
dx x-y+1
(10%) (15) y' + 2xy = exp(x^2) y^2
(二) (10%)
One homogeneous solution of the equation
(1-x^2) y" - 2x y' + 2y = 6(1-x^2)
is y=x. Find the complete solution.
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