課程名稱︰微分方程 課程性質︰系定必修 課程教師︰張帆人 管傑雄 丁建均 黃天偉 開課學院：電資學院 開課系所︰電機系 考試日期（年月日）︰2011/1/12 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 : 1.(14 scores) Find the general solutions of d 運算子(──)=D dt ╭ D˙x(t)- D˙y(t)=0 │ (a) │ │ ╰ D˙x(t)+ D˙y(t)=3[-x(t)+y(t)] (7 scores) ╭ D˙x1(t)- x1(t) = x2(t) │ D˙x2(t) = x2(t) (b) │ D˙x3(t) = x3(t) │ D˙x4(t)- x4(t) = x5(t) │ D˙x5(t) = x5(t) (7 scores) ╰ 2.(16 scores) (a)Find power series solutions about x=0. "Also specify the region of convergence. " (x^2 + 1)y"(x)-2y(x)=0 (8 scores) (b)Find power series solutions about x=0. (x^2)y"(x) + xy'(x) + (x^2-4/9)y(x)=0 (8 scores) 3.(10 scores) Legendre polynomials Pn(x) are orthogonal on the interval[-1,1]. The first 3 Legendre polynomials are P0(x)=1 P1(x)=x P2(x)=(3/2)x^2-(1/2) (a) Verify {P0(x), P1(x), P2(x)} is an orthogonal set. (5 scores) (b) Find the first three coefficients, a0, a1, a2 in the orthogonal series expansion of f(x). f(x)=a0P0(x)+ a1P1(x) +a2P2(x) +.... where f(x)=╭ -1, -1<x<0 │ ╰ 1, 0<x<1 (5 scores) 4.(10 scores) Let Φn(x)≡e^(inπx/p), where i=√(-1), n are integers. Define the set S as S≡{...Φ-n(x), ...Φ-2(x),Φ-1(x),Φ0(x),Φ1(x),Φ2(x),...,Φn(x),...} The inner product of Φm(x) and Φl(x) over [-p,p] is defined as p (Φm(x),Φl(x))=∫ Φm*(x)Φl(x)dx -p where Φm*(x) denotes the complex conjugate of Φm(x). (a) Show that S is an orthogonal set. (5 scores) (b) Assume f(x) is a piecewise continious real function over[-p,p].The complex form Fourier series of f(x) is ∞ Σ Cn e^(inπx/p). n=-∞ Show that 1 p Cn=──∫ f(x)e^(inπx/p)dx. 2p -p (5 scores) (φ^2)u(x,t) (φ^2)u(x,t) 5.(15 scores)Solve 16────── = ────── for π>x>0, t>0 with φx^2 φt^2 (註：φ是partial偏微分) ╭ u(0,t)=0, u(π,t)=0 │ u(x,0)=sinx+2sin2x │ │ φu(x,t)| │ ────| =0 ╰ φt |t=0 (a) Find u(x,t) by the method of separation of variables. (10 scores) (b) Show that u(x,t) can be expressed as (1/2)[f(x+at) + f(x-at)]. (5 scores) 6.(5 scores) (a) Expand f(x)=e^x, 0<x<1 in a cosine series. (3 scores) (b) Find the value of f(999) in problem 6(a). (2 scores) 7.(10 scores) Using Fourier series to solve the following DE. x" + 10x = f(t), x(0)=0, x'(0)=0 f(t)=╭5, 0<t<π f(t+2π)=f(t) ╰0, π<t<2π 8.(10 scores) (a) Find the corresponding time-domain waveform f(t) of the S-domain function e^s 1 ──(───), and graph it. s 1-e^-s (2 scores) (b) FInd the corresponding time-domain waveform f(t) of the S-domain function e^s 1 ──(───), and graph it. s 1+e^-s (2 scores) (c) Use the f(t) of Problem 8(b) and Laplace Transform to solve the DE: x"+ 2x'+ x = 5f(t) , x(0) = x'(0) = 0 (4 scores) (d) Discuss the differences of responses from the Fourier Transform in "Problem7" and the Laplace Transform in Problem 8(c). (2 scores) 9.(10 scores) Please use the following initial value problem (t^2)y" + (t^2)y' +(t^2)y = 0, y(0) = 1, y'(0) = 0 to show the inverse Laplace Transform of 1/√(s^2+1) is Bessel Function: -1 1 L {─────} =J0(t) √(s^2+1) --

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