課程名稱︰ 工程數學
課程性質︰ 系定必修
課程教師︰ 邱文英
開課學院： 理學院
開課系所︰ 化工系
考試日期（年月日）︰ 97/11/5
考試時限（分鐘）： 100分鐘
是否需發放獎勵金： 是
試題 :
1.( 8%) Existence
(a) Dose the ODE y'^(2)=-1 have a "real solution" ?
(b) Dose the ODE │y'│+││y│= 0 have a "general solution"?
2.(48%) Find a general solution
1
(a) xy'= ── y^(2)+y
2
(b) (e^(x+y)+ye^(y))dx + (xe^(y)-1)dy = 0
(c) (x^(2)+y^(2))dx - 2xydy = 0
(d) y'+(x+1)y=e^(x^(2))y^(3)
(e) x^(2)y''+xy'-4y=0
1
(f) y''+9y=cosx+──cos3x
3
(g) y" - 2y + y =e^(x)sinx
1 3
(h) x^(2)y" + xy' - ──y = ── +3x
4 x
3.( 8%) If y1 and y2 are solution of y1' + py1 = r1 and y2' + py2 = r2
respectively, what can you say about the sum y1+y2 ?
4.( 8%) Find the solution of
2(y')^(2) - (2y^(2) + x)y' + xy^(2) = 0,
How could you find a singular solution from general solution?
5.(10%) Find the general solution for x(t) and y(t)
┌ dx
│ ── - 3x -6y = t^(2)
│ dt
│
│ dy dx
│ ── + ── - 3y = e'
└ dt dt
6.(10%) Prove that
1
(a) y = ───e^(ax)
Φ(D)
e^(ax)
= ─── if Φ(a)≠ 0
Φ(a)
1
(b) y = ───e^(ax)
Φ(D)
xe^(ax) dΦ
= ─── if Φ(a) = 0, ───│ ≠ 0
dΦ dD D = a
──│
dD D = a
where Φ(D) = anD^(n) + a(n-1)D^(n-1) + ...... + a1D + ao, and an, a(n-1)...a
& a are constants.
7.( 8%) (Spring) Find the frequency of vibration of a ball of mass m=3kg on a
spring of modulus
(a) k1=27nt/m
(b) k2=75nt/m
(c) on there springs in parallel
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