Laplace transform questions- 'A' grade

PRACTICE PROBLEMS CHAPTER 6 AND 7

I. Laplace Transform

1. Find the Laplace transform of the following functions.

(a) f t =sin 2 t cos 2t 

(b) f t =cos2 3 t 

(c) f t =t e 2t sin 3t 

(d) f t =t 3u 7 t 

(e) f t =t 2 u 3 t 


(f) f t = { 1,
2
if
t −4t 4, if
0≤t2,
t≥2


(g) {
f t = t , if
5, if
0≤t 3,
t ≥3



{
0, if t  ,
(h) f t = t− , if ≤t2
0, if t≥2


(i) f t = {
cos  t  , if
0, if
t4,
t≥4


(j) f t = {
t , if
t
e , if
0≤t 1,
t ≥1


2. Find the inverse Laplace Transform:

1
(a) F  s=
 s1 s2 −1
2 s3
(b) F  s= 2
s 4 s13
e−3s
(c) F  s=
s−2
1e−2 s
(d) F  s= 2
s 6

, 1−e−2 s
3. The transform of the solution to a certain differential equation is given by X s= .
s 21
Determine the solution x(t) of the differential equation.

4. Suppose that the function y t  satisfies the DE y ' ' −2 y ' − y=1, with initial values,
y 0=−1, y ' 0=1. Find the Laplace transform of y t 


5. Consider the following IVP: y ' ' −3 y '−10 y=1, y 0=−1, y ' 0=2
(a) Find the Laplace transform of the solution y(t).
(b) Find the solution y(t) by inverting the transform.

6. Consider the following IVP: y ' ' 4 y=4 u 5 t  , y 0=0, y ' 0=1
(a) Find the Laplace transform of the solution y(t).
(b) Find the solution y(t) by inverting the transform.

7. A mass m =1 is attached to a spring with constant k =5 and damping constant c = 2. At the instant t=
the mass is struck with a hammer, providing an impulse p = 10. Also, x 0 =0 and x'(0)=0.
a) Write the differential equation governing the motion of the mass.
b) Find the Laplace transform of the solution x(t).
c) Apply the inverse Laplace transform to find the solution.


II. Linear systems


1. Verify that x=e 
t 1
0
2 t e t 1
1  is a solution of the system    
x ' = 2 −1 xe t 1
3 −2 −1

2. Given the system x ' =t x− ye t z , y '=2 xt 2 y− z , z ' =e−t 3 t yt 3 z , define x, P(t) and
f t  such that the system is represented as x ' =Pt xf t 

3. Consider the second order initial value problem: u ' '2u '2u=3sin t , u0=2,u ' 0=−1
Change the IVP into a first-order initial value system and write the resulting system in matrix form.


4. Are the vectors x 1 =
  
1
−1 ,
1
0
x2= 1
1
and x 3 =

1
1
1
linearly independent?



5. Consider the system x ' = 
−2 −6 x
0 1 
Two solutions of the system are x 1=e  
t −2

1
and x 2 =e 
−2t 1

0
(a) Use the Wronskian to verify that the two solutions are linearly independent.

(b) Write the general solution of the system.