110年 - 110 國立中央大學_碩士班招生考試_機械工程學/系統組(一般生):工程數學(含程式設計)#105893

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

(b) (5%) Find the solution forx²y"+2xy-6y=0,水(1)=0.5,y'(1)=1.5

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

(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 RIC-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 , and across a capacitor of capactiance C is Q/C, where Q is the charge and the relation between Q(t) and i(t) is Q(t) = si(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=, P and D matrix can be obtained from eigenvectors and eigenvalues of matrix A)

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

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

(b) (4%) Find the values of ao, a1, az, az, b1, bz, bz.

(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)(10%)數學中常用的π可用下列無窮數列來計算:
吾人可藉由計算數列的一項、二項、三項和...,來求得π的近似值。試寫一程式,以迴圈方式計算至n項和之九的近似值,並將結果存於變resul 中。程式碼限定以C、C++、VisualBasic或Fortran 撰寫,除迴圈整數宣告外,其餘所有變數均以實數宣告,並在答案卷上註明使用的程式語言。

(b)(15%)質數是指一個只能被1和自己整除的整数。埃拉托斯特尼篩法是一種找出質數的方法。下列以程式語言來解釋其概念:

1.產生一個陣列,把所有元素的初始值均設為1(真)。之後下標(Index) 確定為質數的陣列元素值保持為1,所有其他不是質數的陣列元素最終會被設為0

2.由陣列下標2開始(因為1不是質數),每一次找到值為1的陣列元素,便往陣列的後方操作,只要下一個下標是目前下標的倍數,便將其所對應的陣列元素設定為0。例如:有一陣列Prime,對於陣列下標2,要讓所有2的倍數的陣列元素值為0,即 Prime[4]=0、 Prime[6]=0、Prime[8]-0, 對於陣列下標3,要讓所有是3的倍數的陣列元素值為0,即Prime[6]=0、Prime[9]=0、Prime[12]=0,

3.當上述程序結束之後,陣列元素中仍然為1者,表示其下標值是個質數。

請撰寫一個程式,使用500個元素的列判斷質數,最後在螢幕上顯示 1~499之間的質數,其中元素0不要使用。不需實際算出,只要寫出程式碼,限定以C.C++、VisualBasic或Fortran撰寫,所有數均以整數宣告, 並在答案卷上註明使用的程式語言。

110年 - 110 國立中央大學_碩士班招生考試_機械工程學/系統組(一般生):工程數學(含程式設計)#105893

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