105年 - 105 國立中山大學_碩士班招生考試_資工系(乙組)：工程數學#105821

1.1 Please find the system of the differential equation in matrix form.

1.2 Please find the characteristic polynomial of the coefficient matrix from 1.1.

1.3 Please find the eigenvalues and associated eigenvectors from 1.1 and 1.2.

1.4 Please find the general solution in matrix form.

1.5 Please find the amount of salt in each tank at time t.

2. The system shown in Figure 2 can be represented by the following two equations.

Solve this system by the Laplace transform and the Inverse Laplace transform, where R = 1 Q, L= 2 H,C=0.5F,v(t)=V,i₁(0)=0,i₂(0)=2A.

3.1 Find the DC value ao and other Fourier coefficients for k ≠ 0 in the Fourier series representation of x(t).

3.2 Define a new signal as y(t) = 2x(t- T₀/2). Use the linearity and time shifing property to write down the Fourier series coefficient for k ≠ 0 for the periodic signal y(r) without evaluating any integrals.

4.1 Determnine a simple expression for A' and show the detail of your calculations.

4.2 Determine a simple expression for As and show the detail of your calculations.

4.3 Conjecture the form of a simple expression for A', k is a posilive integer.

4.4 Prove or disprove your conjecture in 4.3.

5. In a bimolecular rcaction A + B -→ M, a moles per liter of a substance A and b molcs per liter of
a substance B are combined. Under constant temperature the rate of reaction is
y'=k(a-y)(b-y)
, that is, y' is proportional to the product of the concentration of the substances that are rcacting, wherd
P(r) is the number of moles per liter which have reacted after time 1. Solve this ODEi. assuming that utb.

6. Find the gencral solution of the Tollowing cquation and show the detail of your calculations.

(4x²D²-24xD+49I)y=36x⁵

105年 - 105 國立中山大學_碩士班招生考試_資工系(乙組)：工程數學#105821

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