110年 - 110 國立中央大學_碩士班招生考試_機械工程學系、機械工程學系(光機電工程)、能源工程研究所_(共同)：工程數學#125285

(a) (5%) Find the solution for y"-3y'-4y = 0, y(0) = 2, y'(0) = 1

(b) (5%) Find the solution for x²y"+2xy'−6y = 0, y(1) = 0.5, y'(1) = 1.5

(c) (5%) Find a basis of solutions by the Frobenius method of the following ODE:
(x+1)² y" + (x+1)y' – y = 0.

(a) (5%) Based on this law, model the current, i(t), for the circuit shown in the following figure.

(b) (5%) Obtain the "steady-state" current in the RLC-circuit when R=50 Ω (Ohm),L = 30 H (Henry), C = 0.025 F (Farad), and E = 200 sin(4t) V (Volt)

Hint: The voltage drop for a current i(t) across a resistor of resistance R is Ri(t), across an inductor of inductance L is L, and across a capacitor of capacitance C is Q/C, where Q is the charge and the relation between Q(t) and i(t) is Q(t) = ∫ i(t)dt

3. (5%) Determine the existence and uniqueness of the solutions to the system

4. (5%) The traffic flow problem can be described by the following table. Please determine the
general flow pattern for the network

(a) (3%) Let A =. Are u and v eigenvectors of A?

(b) (4%) Show that 7 is an eigenvalue of matrix A in (a), and find the corresponding eigenvectors.

(c) (8%) Find a formula for A^k, k≥ 1 (Hint: given that A= PDP^-1, P and D matrix can be obtained from eigenvectors and eigenvalues of matrix A)

6. (10%) Let f(t) be a periodic function, f(t) = f (t + p) with period p. Denote L[f(t)] as the Laplace transform of f(t). Prove L[f(t)] =.

(a) (3%) Find the (fundamental) period T of f(t).

(b) (4%) Find the values of a₀, a₁, a₂, a₃, b₁, b₂, b₃.

(c) (3%) Will the Fourier series converge to f(t)? Explain your reasons within 30 words.

(d) (5%) Can one obtain identical Fourier series of any function g(t) by using 2T and T in Equation (1)? Explain your reasons within 30 words.

(a) (6%), (6%) Please first prove Equations (2) and (3), respectively

(b) (6%), (7%) Please give their (Equations (2) and (3)) matching engineering (or physical) examples with discussion, respectively.