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

(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

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 Ak, 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).

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