課程名稱︰工程數學一
課程性質︰化工系大二必修
課程教師︰李克強
開課學院：工學院
開課系所︰化工系
考試日期（年月日）︰2020年1月10日
考試時限（分鐘）：130分鐘
是否需發放獎勵金：是
試題 : (試題最後面有給一堆Laplace轉換公式還有J0(x)和Y0(x)的表格，不怕寫不出來)
(一)Use Laplace transform to solve the ODE-IVPs
(A)y"+4y=δ(t)+u(t-2) ；y(0)=0, y'(0)=1 10%
(B)y"+y=sin(t)-u(t-4π)sin(t-4π) ；y(0)=0, y'(0)=0 10%
(C)y"+2y'+y=sint+δ(t-π) ；y(0)=0, y'(0)=0 10%
(D)Use the convolution theorem to solve the following ODE-IVP 10%
y"+3y'+2y=cost ；y(0)=1,y'(0)=0
(二)Find the first three non-zero terms of each of the two linearly-indepedent
series solutions of the following ODE
y"-2xy'+10y=0,-∞<x<∞ 20%
(三)Find the solution of the follow ODE
x^2*y"+xy'+4x^2*y=0
subject to the following boundary conditions
y(1/2)=1, y(1)=2 and 1/2=<x=<1
What is the value of y(3/4)? (取兩位有效數字或以符號表示) 15%
(四)
(A)Show that x=0 is a regular point of the ODE
xy"+2xy'+6e^xy=0
and find the two roots of the indical equation 5%
∞
(B)Solve for the Frobenius solution y(x)=Σam*x^(m+r) corresponding to the
"Larger root". Obtain the first three non-zero term 10%
m=0
(五)Determine the general solution of the following ODE that is vaild in any
inerval not including the singular point 10%
(x-2)^2*y"+5(x-2)y'+8y=0
(Hint:you can also try t=x-2 if you like)