課程名稱︰常微分方程導論
課程性質︰數學系大二必修
課程教師︰陳俊全
開課學院：理學院
開課系所︰數學系
考試日期︰2018年01月12日(五)
考試時限：08:10-10:00，計110分鐘
試題 :
Choose 4 from the following 6 problems.
(4)
1. Solve the equation y + 2y'' + y = cos t.
2. Use the Laplace transform to solve the equation
y''' - y'' + y' - y = 0, 0≦t≦5
t, 5＜t
y(0) = 0, y'(0) = 0, y''(0) = 1.
3. Given that y1(t) = t is a solution to
2
(1-t ) y'' + 2ty' - 2y = 0, -1 < t < 1,
use the method of reduction of order to solve the equation.
4. For a matrix A=(a_ij)_(m x n), let ||A|| = (sum a_ij^2)^(1/2).
At ||A||t
(a) Show that ||e || ≦ e for t≧0.
At At
(b) Let det e denote the determinant of e .
At
Show that det e = exp{(sum a_ii)t}.
(c) Let Z(t) be a fundamental matrix of the equation x'=Ax. Prove that
At -1
e = Z(t)Z (0).
5. Let
( 0 1 0 )
A = ( 0 0 1 ) .
(-1 -3 -3 )
At
Find e and solve the initial value problem
( 0 ) ( 1 )
x'(t) = Ax(t) + ( 1 ), x(0) = ( 0 ).
( t ) ( 0 )
6. Find all the critical points for the following system and discuss the
stability of the critical points.
dx dy 3