108年 - 108 國立成功大學_碩士班招生考試_電機工程學系、電機資訊學院(微電、奈米聯招):工程數學#106070

1.20% Let matrix

Find its eigenvalues and eigenvectors and write the vector

as a combination of those eigenvectors.
(a) Solve the equation du/dt = Au starting with the same vector u(0) at time t = 0.

2. 10% Let
Write the Laurent series expansion of f(z) for 5 <| . z -2 /<∞

(a) Let a and bin equation (1) are given constants, what wil the equation be transformed to if

z = In x is substituted in it.

x²y "+ axy't by = 0 (x>0) (1)

(b) Use the result of part (a) to find the general solution of
x²y"-3xy'+ 3y =0

4. 15% Solve the following differential equation:
(x- 4x²y³)dy+(4x⁴-y)dx=0

5. 15% Solve the following differential equation, using Laplace transformation method. y(t) is a function oft and u(t) is the unit step function.
y" + 2y = y(0)=1, y'(0)=0

6. 20% Find the general solution of the following equation:

108年 - 108 國立成功大學_碩士班招生考試_電機工程學系、電機資訊學院(微電、奈米聯招):工程數學#106070

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