課程名稱︰工程數學二
課程性質︰土木系必修
課程教師︰劉格非
開課學院：工學院
開課系所︰土木系
考試日期（年月日）︰101/05/06
考試時限（分鐘）：150分鐘
是否需發放獎勵金：需要
（如未明確表示，則不予發放）
試題 :
Problem (1)
y''' - 3y'' + 3y' - y = 24t*exp(t) + ln(t) (y為t的函數 y'=dy/dt)
(a) (9%) Find the homogeneous solution.
(b) (4%) Find the particular solution corresponding to 24t*exp(t).
(c) (9%) Find the particular solution corresponding to ln(t).
t
Your result should be expressed in terms of G(t) =∫ln(x)*exp(-x)dx.
0
(d) (4%) From y(0)=1, solve one of the integration constant.
Problem (2)
Use Frobenius series to solve the following equation with the center x = 1:
(x-1)* y'' + 2(x-1)* y' + 2(1-x)* y = 4(x-1) y(1) = 1, y'(1)=0
(a) (4%) Find the indicial equation and solve r.
(b) (8%) For the smaller number of r, find the series expansion solution with
respect to x=1. This solution is y1.
(c) (4%) Identify the above solution which part is homogeneous solution and
which part is the particular solution yp.
(d) (6%) Use y1 - yp as one of the homogeneous solution and find the other
homogeneous solution. (If you cannot finish the integration, just
leave it in integration form, but make sure you have make the
solution compact. )
(e) (6%) If y(1)=1, y'(1)=0, write down the first 3 terms for the solution.
(Hint: It will be fast if you transfer x coordinate first.)
Problem (3)
(16%) Solve t* y'' + 2(1-t)* y' - 2y = t* exp(-t) y(0)=0 , y'(0)=1
Problem (4)
___
(a) (5%) Prove Γ(1/2) = √π
______
(b) (5%) Prove J = √2/πx *sin(x)
1/2
(c) (5%) Write down the general solution for
(x^2)* y'' + (4x)* y' + (4x^2 - 9v^2)* y = 0 , 0≦x≦5
Then with the boundary condition y(0)=0, write down the solution.
Explain why we need only one boundary condition.
exp(-π)
(d) (5%) Find the inverse Laplace transform of _______________ .
2 2
(s+π) - π
Problem (5)
(10%) Prove the Laplace transform of f(x)*g(x) is just the multiplication of
the Laplace transform of f(x) and Laplace transform of g(x).
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