課程名稱︰工程數學一
課程性質︰系必修
課程教師︰王勝仕
開課學院：工學院
開課系所︰化工系
考試日期（年月日）︰2010年11月5日
考試時限（分鐘）：140分鐘
是否需發放獎勵金：是
試題 :
1.Basic concepts (16/110)
Please describe the the properties, including (A)linear or non-linear
(B)homogeneous or non-homogeneous (C)order (D)degree of the given ODEs.
(a)y(y")^3+5(cotx)y=cos2x
(b)e^(-tanx)y"-(coshx)y'+5xy=ln(e^x)
2.(a)(6/110)
(1)Show the Wronskian determinant of {logx^2,logx^3,logx^4}
(2)Determine whether logx^2,logx^3,logx^4 are linearly indenpendent or
linearly dependent?
(b)Please show the linearly dependence of cos(2lnx),sin(2lnx) and find the
ODE for which cos(2lnx),sin(2lnx) are solutions.
3.(15/110)Given a first order ODE: y'+J(x)y=K(x)
(a)Please use two different ways to derive the general solution of the ODE.
(b)What are the homogeneous solution and particular solution?
(c)Find the general solution of x^3y'+3x^2y=5sinh10x
4.(48/110)Please solve the following ODEs:
(a)y'=(3x-y-9)/(x+y+1)
(b)(2x*y^2+y)dx+(x+2x*y^2-x^4*y^3)dy=0
(c)[(-y/x)+2cos2x]dx+[(1/x)-2sin2y]dy=0
(d)xy'=(y/x)^3+y
(e)2yy'+y^2*sinx=sinx,y(0)=2^(1/2)
(f)(sinycosy+x(cosy)^2)dx+xdy=0
(g)4y^4+3cosx+(16y^3*x-3cosy)y'=0
(h)y'=-2-y+y^2
5.(15/110)Two-tank mixing process
(a)Please write down the general "mass conservation equation" and show the
unit for each term using gal,lb and min as the units of volume,mass and
time.
(b)Please write down the "mass conservation equation" for each tank using the
symbols(Q1,Q2,Q3,V1,V2,d0,d1 and d2) shown on the board.
(c)Tank 1 initially contains 20 lb of salt dissloved in 100gal of brine while
tank 2 contains 150 gal of brine which 90 lb of salt are dissloved. At
time 0, a brine solution containing 0.5 lb of salt per gal is added to
tank 1at the rate of 5 gal/min. Tank 1 has an output that discharges brine
into tank 2 at the rate of 5 gal/min, and tank 2also has an output of 5
gal/min.Given the specific values of all flow stream parameters/variables
described above, please show the amounts of salt in tank 1 and tank 2 at
any time t, name A1(t) and A2(t), respectively.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.245.85
※ 編輯: mgla8063 來自: 140.112.245.85 (11/05 19:18)