[試題] 99上 陸駿逸 化學數學一 期末考
課程名稱︰化學數學一
課程性質︰化學系大二必修
課程教師︰陸駿逸副教授
開課學院:理學院
開課系所︰化學系
考試日期(年月日)︰2011/01/12
考試時限(分鐘):10:20~12:10 然後再延長到12:40
是否需發放獎勵金:是
(如未明確表示,則不予發放)
試題 :
Chemical Mathematics Final Examination
2011/01/12 10:20~12:10
1(20pts) Solve the functions y(x) in the following ODEs
' 2
a.y = -2y
'
b.y -xy = x
" '
c.y + 5y +4 = 0
" '
d.y + 5y +4 = 3sin(2x)
2(20pts) Find the z component of r ×▽ in terms of the cylindrical coordinate.
→ →
Hint: r= r r + z z
3(25pts) Use the method of seperation variables to solve the function f, which
obeys the 1D diffusion equation. 2
δ δ
____ f(x,t) = ____ f(x,t)
δt δx^2
within the range[0,π], and the bounbary conditions:
δ
f(x=0,t) = 0, ____f(x=π,t) = 0, f(x,t=0) = x(2π-x)
δx
Hint:
2 2
π 2-(2+n π )cosnx
∫ [x(2π-x)sin(nx)]dx = ______________________
0 3
n
(δ是偏微分)
4(20pts) On the(x,y) plane, use the variational method to find the shape of a
curve of minimum length which incloses the area π.(Hint: use the polar
coordinate.)
5(25pts) Using the series method, find suitable k so that the solution of the
following ODE does not diverge within the range (-∞,∞). Obtain (any) three
non-zero solutions.
" '
y - 2xy + 2ky =0
6(20pts) Given the generating function of the Hermite polynomials
2 n
2xh-h ∞ h
φ(x,h)= e = Σ Hn(x) __
n=o n!
show the recursive relation
2xHn(x) = Hn+1(x) + 2nHn-1(x)
n
(Hint: Differential φ(x,h) with respect to h, and equate coefficients of h )
心得:鬼才會寫..............
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