推 a22525740 : 國務院總理推 01/23 01:01
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）︰2015/01/16
考試時限（分鐘）：110（後來延長至120min）
試題 :
1.(40%) Find the first three terms in each of the two linearly independent
power series solutions in powers of x of
d^2 y
------- + (e^x)y = x
dx^2
2.(25%) Find the first three eigenvalues and corresponding eigenfuctions
of the problem:
d^2 y dy
(x^2)------- + x---- + (λ^2)(x^2)y = 0
dx^2 dx
y(0) = finite, y'(1) = 0, x 屬於 [0,1]
3.(25%) Find the eigenvalues and corresponding eigenfuctions of the
boundary value problem:
d^2 y
------- - λy = 0 , y(0) = 0, y'(1) = 0
dx^2
(Assume that all eigenvalues are real, i.e., λ屬於實數)
4.(10%) Use Newton-Raphson method to solve the algebraic equation
numerically (error=10^(-4)):
x^2 - 1 = sin x
How many solutions are there?
附：
J(x)的公式
J0(x) J1(x) Y0(x) Y1(x) 的查表
J0(x)=0 for x=2.4048, 5.5201, 8.6537, 11.7915
J0(x)=0 for x=0, 3.8317, 7.0156, 10.1735, 13.3237
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