1. Application for Second-order Differential Equation (30%) A simplified circuit model for transmission line is as follows. Suppose we have an al- ternating current input with voltage U; = Ucos(wt), the equivalent inductor, resistance and capacity of the line are noted as L, R.C, the load has a single resistance RL. R U(t) (a) Formulate the differential equation of the output voltage U. beside the load Ry using L, R.C. RL,U.. (b) Find the steady-state solution for the equation. (c) Choose the best value for resistance Ry such that the amplitude of voltage U. is maximized. (The voltage-current relationship of the components are UR = R . IR, UL = Ldi, Io = Clue. 2. Periodic Functions (30%) Find the following functions' minimum positive period. (a) F(x) = cos(1) (b) F(x) = Zin=-20 ( x-n)= (c) F(x) = "' I = $ where p and q are co-prime integers is irrational 3. Orthogonality of Trigonometric functions (20%) (a) Prove the orthogonality of sine functions, which is sin(ma)sin(no) dr = 0,m #n I, m = n3. Orthogonality of Trigonometric functions (20%) (a) Prove the orthogonality of sine functions, which is [ sin(ma)sin(ne)dx = 0, m # n IT, m = n (b) Use the expansion f(x) = do+ 2, (ancos(nr) + basin(nx)) to prove [If(2) Pax = 2mao ? + [(land? + 161?) n=1 4. Fourier Series (20%) Calculate the Fourier series of the following functions. (a) F(x) = [sin(x/2) | with period 2x (b) F(x) = tan(x) with period #, give the first 7 terms (constant and the first 3 sine and cosine terms)