課程名稱︰常微分方程導論
課程性質︰數學系二年級必修
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）︰2014/1/10
考試時限（分鐘）：8：10～10：00
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 7 problems.
(4) t
1. Solve the equation y － 2y'' + y = e + t.
2. Use the Laplace transform to solve the equation
︴2t, 0 ≦ t ＜ 1,
y'' + y = ︴ y(0) = y'(0) = 0.
︴0, 1 ≦ t ＜ ∞,
2
3. Solve the equation (1 - t)y'' + ty' - y = t , 0 < t < 1.
(Hint: y(t) = t is a solution of the homogeneous equation.)
4. Solve the linear system
（ 2 1 1 ） （ 0 ）
x'(t) = （ 2 2 -1 ）x(t), x(0) = （ 1 ）.
（ 0 -1 2 ） （ 0 ）
5. Solve the linear system
2t
( 5 4 ) ( e ) ( 2 )
x'(t) = ( -2 1 ) x(t) + ( 0 ), x(0) = ( 1 ).
6. Let Ψ(t) be a fundamental matrix for the system x'(t) = A(t)x(t). Show
that a solution of the system x'(t) = A(t)x(t) + g(t) has the form
-1 t -1
x(t) = Ψ(t)Ψ (0)x(0)＋Ψ(t)∫Ψ (s)g(s)ds.
0
7. Let A be a constant n ×n matrix. Explain the meaning of exp(A).