[試題] 96上 化學數學一 期中考

[brucepan7782 (bruce)]

課程名稱︰化學數學一
課程性質︰必修
課程教師︰陸駿逸
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰96.11.16
考試時限（分鐘）：150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.(4pts) Find the integration factor to solve the following differential

         equation.   (1/x^2)[dy(x)/dx] +y(x) +exp(x^3) =0

2.(6pts) Use the inverse operator method to obtain the general solution of the

         ODE    (D^2 -5D +6)y(x)=x    D = d/dx

3.(8pts) Derive the Euler equation for the unknown function ρ(x) which

         minimizes the following functional

                 L
         F[ρ] =∫{Tρ(x)lnρ(x) +a[dρ(x)/dx]^2 +b[ρ(x)]^2 +c[ρ(x)]^3} dx
                 0
                                        L
         subjected to the constraint   ∫ρ(x)dx =1
                                        0
         where T, a, b, and care constant

4.(10pts) Use the series solution method to solve the ODE

          (1-x^2)D^2 y(x) -2xDy(x) +6y(x) =0

5.(12pts) Find the general solution of the coupled ODE

          D^2 Y1(x) +Y1(x) -Y2(x) =exp(x)

          D^2 Y2(x) +4Y2(x) -Y1(x) =0

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