Class notes
Laplace transform questions- 'A' grade
- Course
- Institution
All major topics and subtopics around laplace are covered in this document. Solving ode using laplace transforms.
[Show more]
Seller
Follow
msrikanthkarthikeyan
Content preview
PRACTICE PROBLEMS CHAPTER 6 AND 7I. 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 3u 7 t
(e) f t =t 2 u 3 t
(f) f t = { 1,
2
if
t −4t 4, if
0≤t2,
t≥2
(g) {
f t = t , if
5, if
0≤t 3,
t ≥3
{
0, if t ,
(h) f t = t− , if ≤t2
0, if t≥2
(i) f t = {
cos t , if
0, if
t4,
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=
s1 s2 −1
2 s3
(b) F s= 2
s 4 s13
e−3s
(c) F s=
s−2
1e−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 21
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 xe t 1
3 −2 −1
2. Given the system x ' =t x− ye t z , y '=2 xt 2 y− z , z ' =e−t 3 t yt 3 z , define x, P(t) and
f t such that the system is represented as x ' =Pt xf t
3. Consider the second order initial value problem: u ' '2u '2u=3sin t , u0=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.
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller msrikanthkarthikeyan. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $8.69. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75759 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now