推 lovehan:推一下 施文彬 XD 02/08 13:24
課程名稱︰工程數學上
課程性質︰機械系大二上必修
課程教師︰施文彬
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2005.11.14
考試時限（分鐘）：110 min
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Rule: No calculators are allows. You are allowed to bring an A4 size
information sheet. Please provide the details of your calculation.
Good luck!
1. Solve the following differential equations.
x^2 x^2
(a)(7%) e dy +[ye /x -1]dx = 0
y 2
(b)(7%) xe dy +[(x +1)/y]dx = 0, y(1) = 0
2
(c)(8%) x dy/dx = y(x+2y)
2. Consider the differential equation y"+y'+2y =cosx
(a)(5%) Find the solution to the homogeneous equation
(b)(5%) Find a particular solution
(c)(5%) Write down the general solution
(d)(5%) Find the solution that satisfies the initial condition
y(0)=1, y'(0)=2
(e)(8%) Write down the general form of a particular solution to
y"+2y'-3y =4 +7sinx +x^2 +e^x
Do not solve for the (undetermined) coefficients.
3.(a)(15%) Find the first four nonzero terms of two linearly independent
2 2 3
series solution of 3x y"+(6x -7x)y'+3(1+x)y =0
(b)(10%) Find the first four nonzero terms of the series solution of the
given initial value problem, about the point where the initial
conditions are given:
y"-ln(x)y'=-1+x; y(1)=1, y'(1)=π.
4. Find the Laplace transform of
t 2(t-v)
(a)(5%) f(t)=∫ e sinh(v)dv
0
t -3τ
(b)(5%) f(t)=∫ e sin(τ)dτ
0
5.(15%) Use the Laplace transform to solve the initial value problem
2t
y"-2y'-15y = e +2δ(t-1); y(0)=0, y'(0)=0.
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