110年 - 110 國立臺灣大學_碩士班招生考試_數學研究所:常微分方程#100326

(a) (10 points) Calculate, where I is the two by two identity matrix.

(b) (10 points) Solve the differential system

with initial condition X(0) = (1,0)T.

2. (20 points) Solve the differential equation

 
 with initial condition x(0) =x'(0) = 0, x"(0) = 1.

3. (20 points) Solve the differential equation

with initial condition x(0) = 0.

(a) (10 points) Find the value of Φ(π).

(b) (10 points) Let X(t) be any other solution of (0.4). Show that

5. (20 points) Solve the differential equation