課程名稱︰化學數學一 課程性質︰化學系大二必修 課程教師︰陸駿逸副教授 開課學院：理學院 開課系所︰化學系 考試日期（年月日）︰2011/01/12 考試時限（分鐘）：10:20~12:10 然後再延長到12:40 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Chemical Mathematics Final Examination 2011/01/12 10:20~12:10 1(20pts) Solve the functions y(x) in the following ODEs ' 2 a.y = -2y ' b.y -xy = x " ' c.y + 5y +4 = 0 " ' d.y + 5y +4 = 3sin(2x) 2(20pts) Find the z component of r ×▽ in terms of the cylindrical coordinate. → → Hint: r= r r + z z 3(25pts) Use the method of seperation variables to solve the function f, which obeys the 1D diffusion equation. 2 δ δ ____ f(x,t) = ____ f(x,t) δt δx^2 within the range[0,π], and the bounbary conditions: δ f(x=0,t) = 0, ____f(x=π,t) = 0, f(x,t=0) = x(2π-x) δx Hint: 2 2 π 2-(2+n π )cosnx ∫ [x(2π-x)sin(nx)]dx = ______________________ 0 3 n (δ是偏微分) 4(20pts) On the(x,y) plane, use the variational method to find the shape of a curve of minimum length which incloses the area π.(Hint: use the polar coordinate.) 5(25pts) Using the series method, find suitable k so that the solution of the following ODE does not diverge within the range (-∞,∞). Obtain (any) three non-zero solutions. " ' y - 2xy + 2ky =0 6(20pts) Given the generating function of the Hermite polynomials 2 n 2xh-h ∞ h φ(x,h)= e = Σ Hn(x) __ n=o n! show the recursive relation 2xHn(x) = Hn+1(x) + 2nHn-1(x) n (Hint: Differential φ(x,h) with respect to h, and equate coefficients of h ) 心得：鬼才會寫.............. --

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