課程名稱︰常微分方程導論
課程性質︰數學系必修
課程教師︰陳俊全
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2010/01/04
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Choose 4 from the following 6 problems.
(4)
1. Solve the equation y + 2 y'' + y = 3sin3t.
2. Use the Laplace transform to solve the equation
(4)
y - y = δ(t - 2) , y(0) = y'(0) = y''(0) = y'''(0) = 0
3. Use the method of reduction of order to find a second solution of the
eqution
(x - 1)y'' - xy' + y = 0, x > 1 ; y1(x) = expx
4. Find the general solution of the linear system
( -8 -1 0 )
x'(t) = ( 16 0 -1 ) x(t)
( 0 0 1 )
5. Find the solution of the linear system
( 2 1 ) ( t ) ( 1 )
x'(t) = (-5 -2 ) x(t) + ( 3 ), x(0) = ( 2 )
6. Let L{f(t)}(s) denote the Laplace transform of f(t). Suppose L{f(t)}(s) and
L{g(t)}(s) both exist for s > a ≧ 0. Prove that L{h(t)} = L{f(t)}L{g(t)}
for s > a, where t
h(t) = ∫ f(t - σ)g(σ) dσ
0
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