課程名稱︰工程數學-微分方程 課程性質︰系訂必修 課程教師︰林清富 張帆人 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）︰2009.01.14 考試時限（分鐘）：150分鐘 是否需發放獎勵金：是 試題 : 1.(5%)Find a power serier solution of the differential equation y"-4xy'-4y=e^x about x=0. 2.(5%)Is x=o an ordinary or a singular point of the differential equation xy"+(sin x)y=0? 3.(5%)If the Laplace transform of f(t) is F(S), and k>0, then find the Laplace transform of e^at*f(t-k)U(t-k). 4.(10%)Use the Laplace transform of the initial-value problem ty"+y'+ty=0, y(0)=1,y'(0)=0 to show that L{J0(t)}=1/√(s^2+1).(Hint:F(s)=L{F(t)}, f(0)=lim s→∞ sF(s) ) 5.(5%)Use the Laplace transform to solve the following BVP y"+2y'+y=0,y'(0)=2, y(1)=2. -1 6.(5%)Find the following inverse Laplace transform L {8k^3s/(s^2+k^2)^3}. ┌ 0 1 0 ┐ 7.(20%)Let A=│ 0 0 1 │, └ 0 0 0 ┘ ┌ 1 ┐ (a)Solve dX(t)/dt=AX(t) with initial condition X(0)=│ 1 │. └ 1 ┘ (b)Find Φ(t), such that the solution of (a) is X(t)=Φ(t)X(0) where X(0) is the initial condition at t=0. (c)Find Φ(t), such that the solution of (a) is X(t)=Φ(t)X(2) where X(2) is the initial condition at t=2. ┌ 0 ┐ (d)Solve dX(t)/dt=AX(t)+f(t) with initial condition X(0)=│ 0 │ and └ 0 ┘ ┌ 0 ┐ f(t)=│ 0 │. └ sin(t) ┘ 8.(15%) (a)Show that the product of two odd functions is even. T (b)Show that ∫ f(t)=0 if f(t) is an odd function. -T (c)Let f(t)=e^-t for 0≦t≦1. Please sketch the plots of the associated half range expansions of sine series, cosine and Fourier series for -2≦t≦2; respectively. 9.(15%) (a)(8)The wave equation a^2Exx=Ett, 0<x<L, t>0 is subject to the initial and boundary conditions: E(0,t)=E(L,t)=0, t>0 E(x,0)=f(x), Et(t=0)=0, 0<x<L Please show that the solution of the wave equation can be written as E(x,t)=0.5[f(x+at)+f(x-at)] [Hint:use the identity 2sinαcosβ=sin(α+β)+sin(α-β)] (b)(7)If the wave equation is define for the infinite region -∞<x<∞ and t>0 with the initial and boundary conditions removed, then please give a procedure that will lead to the general solution of the wave equation with the form E(x)=F(x+at)+G(x-at) where F and G are two arbitrary twice-differentiable function. 10.(15%)Use the result that Fourier transform of e^(-x^2/4p^2) is 2√πpe^(-p^2α^2) to solve the following equations (a)(7)Ut=DUxx, -∞<x<∞, t>0 U(x,0)=δ(x), -∞<x<∞. (b)(8)Ut=-U/τ-vUx+DUxx, -∞<x<∞ t>0 U(x,0)=δ(x), -∞<x<∞. --

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