課程名稱︰工程數學二
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）︰2011年4月29日
考試時限（分鐘）：140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.The ODE is given as follows:
x^2y"-4xy'+(x^2+6y)=0
(a)Which type of the power series solution, Taylor series solution or
Frobenius series solution, exists at x=0 for the above ODE? Explain why?
(b)What are the corresponding indicial equation and recurrence formula/
relation?
(c)Find the series solution of the above ODE.
2.Consider a set of linearly independent (but not orthogonal) functions:
{v1,v2,v3}, x=[-1,1].
(a)How can you transform the above linearly independent functions into an
orthogonal function set {P1,P2,P3}, x=[-1,1]? Show the formula you would
use in the transformation process (v1,v2,v3與P1,P2,P3間的關係?)
(b)If {v1,v2,v3}={1,x,x^2}, x=[-1,1], please show that these three functions
are linearly independent.
(c)If {v1,v2,v3}={1,x,x^2}, x=[-1,1], what are the norm of v2? What are P2
and P3?
(d)Given that f(x)=e^x=c1P1+c2P2+c3P3, x=[-1,1], find the values of the
constants c1,c2 and c3.
3.Considering the following questions
(a)Find the general solution of the question:
x^2y"-5xy'+9(x^6-8)y=0
(b)Plot the graphs of Y0(x), J0(x) and J1(x) and also show the values of
J0(0), J1(0), and Y0(0).
(c)Write done the Bessel basic formula, and use Bessel basic formula to
obtain Bessel Recurrence Formula and Bessel differential formula.
(d)Use J0(x)=A and/or J1(x)=B to express J4(x)and J'0(x).
4.Considering the following questions:
(a)Is {sin(nx)} a complete orthogonal set? What would be the corresponding
Sturm-Liouville BVP system of the above function set?
(b)Why do we have to learn Sturm-Liouville BVP? How many non-trivial
solution (or eigenfunctions) can we obtain from Sturm-Liouville BVP? What
are the key properties for the eigenfunctions and eigenvalues obtained
from Sturm-Liouville BVP? What purposes/functions do these eugenfunctions
serve?
(c)In general, we can expand a function using the generalized Fourier series
and Tylor series. What are the differences between these two series
expansions?
(d)Can the equation (*) be transformed into Sturm-Liouvile type equation?
How? What is the type of the boundary condition (**)? Find the eigenvalues
and corresponding eigenfunctions of the above Sturm-Liouville BVP system
(*) + (**).
y"-2y'+(λ+1)y=0 (*)
y(0)=y(4)=0 (**)
Please also show that any two different eigenfunctions (with different
eigenvalues) are orthogonal.
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※ 編輯: mgla8063 來自: 140.112.245.85 (06/25 00:39)