- 1 1
- 1 - 1.Find the general solution for each of the following differential equations.
(a). 2 23
2 1
3 1
x xy
y x y dx
dy
−
− +
= + (10%) (b). y′′+ y′− 2y =5t+e2t, y(0)= y′(0)=1 (10%) 2.Use the Laplace transforms to solve the given integral equation. (10%)
−
− −
−
= e t te t y d
t
y 0
)
( ( )
3 2 )
( τ τ τ
3.Use the Laplace transforms to solve the following equation. (10%)
>
≤
= ≤
′=
′′+
2 , 5
2 0 , ) 0
( , )
( t
t t g t
g y
y , y(0) = y′(0) = 0
4.Find the inverse Laplace transforms of the following equation. (10%)
) 26 2 )(
2 ( ) 1
( 2
+ + +
= +
s s s s s F
5. (a). Find the Fourier series of the function f(x),
where f )(x = x+π if −π <x<π and f(x+2π)= f(x). (10%) (b).Using (a) to evaluate − + − +−
7 1 5 1 3
1 1 ? (5%)
6. (a). Find the eigenvalues and eigenvectors of the matrix
−
= −
4 3
2
A 1 . (10%)
(b). Using Cayley-Hamilton Theorem to evaluate A . (5%) 20
7. (a). Show that the function )
1 tan( 1 )
( = +
z z
f there is infinitely many singularities, only one of which is nonisolated. (5%)
(b) Evaluate dz
z z z
z
C − −
+ ) 7 )(
(
3
π , the contours C consists of the circle z =6, described in the
positive direction, together with the circle z =4, described in the negative direction.
(5%)
8. Find the Cauchy principal value of the integral dx
x I ∞ x
∞
− + −
= ( 1)( 1)
3
2 . (10%)
