[試題] 101-1 夏俊雄 常微分方程導論 期末考

[Malzahar (虛空先知)]

課程名稱︰常微分方程導論
課程性質︰必修
課程教師︰夏俊雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2012/12/28
考試時限（分鐘）：100分鐘

試題 :

                            ODE ExAm 3 12/28/2012

In this exam, let f(t) and g(t) be piecewise continuous function defined on
[0,∞). The convolution of f(t) and g(t) are defined as

                                        t
                           (f*g)(t) := ∫ f(t-s)g(s)ds.
                                        0

1. (10 points) Let L denote the Laplace transform and L^-1 denote the inverse
Laplace transform. Show that

                       L(f*g)(s) = L(f)(s)L(g)(s).

2. (10 points) Solve the integro-differential equation

                            t
               y'(t) = 1 - ∫ y(t - s)e^-3s ds,     y(0) = 2.
                            0

3. (10 points) Do you think the following equation have periodic solutions?
Prove or disprove it.

                     x'' + x + x^3 = 0.

4. (80 points) Find general solution for the following differential equations.

               (1)   (t^2)x'' - 3tx' + 4x = 0,    t>0,
               (2)   x'''(t) - 3x''(t) + 3x'(t) - x(t) = 2t + e^t,    t∈R,
               (3)   x''(t) + tx'(t) + 2x(t) = 0,      t∈R,
               (4)   y'' + y = u_4π(t),    t≧0.

5. (10 points) Show that for certain μ∈R, the van der Pol equation

               u'' - μ(1 - u^2)u' + u = 0

has time periodic solutions. What theorem shall you apply?


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※ 編輯: Malzahar (118.166.208.171), 02/12/2015 15:34:17
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