Tuesday, November 10, 2015

Advanced Math Solutions – Laplace Calculator, Laplace Transform

In previous posts, we talked about the four types of ODE - linear first order, separable, Bernoulli, and exact. In today’s post, we will learn about Laplace Transforms, how to compute Laplace transforms and inverse Laplace transforms. I’ll admit I was afraid of Laplace transforms before I learned it. The new symbols and the messiness of the problem really intimidated me. However, once I learned what Laplace transforms are and how to do these types of problems, I began to realize that it isn’t so scary. With the assistance of a table and some formulas, anyone can do Laplace transforms.

What is a Laplace Transform?

Laplace transforms can be used to solve differential equations. They turn differential equations into algebraic problems.

Definition:
Suppose f(t) is a piecewise continuous function, a function made up of a finite number of continuous pieces. The Laplace transform of f(t) is denoted L{f(t)} and defined as:
L\left\{f(t)\right\}=\int_{0}^{\infty}e^{-st}f(t)dt
If you see L\left\{f(t)\right\}=\int_{-\infty}^{\infty}e^{-st}f(t)dt, then you can assume that for t\lt0,\quad f(t)=0, and then you can use the original definition of Laplace transform.


Now, we will get into how to compute Laplace transforms:

Laplace transforms can be computed using a table and the linearity property, “Given f(t) and g(t) then, L\left\{af(t)+bg(t)\right\}=aF(s)+bG(s).” The statement means that after you’ve taken the transform of the individual functions, then you can add back any constants and add or subtract the results.

Look at the table and see what functions you can transform. Algebraic manipulation may be required. The table will be your savior when it comes to these problems.



Simple enough! Let’s look at an example (click here):



This problem is very simple. It requires looking at the different functions, finding the corresponding transforms in the table, and then adding any constants and adding and subtracting the results together.


Here’s another example (click here):




Let’s get into inverse Laplace transforms!

What is an inverse Laplace transform?

An inverse Laplace transform is when we are given a transform, F(s), and asked what function(s) we had originally.
Definition:
f(t)=L^{-1}\left\{F(s)\right\}

How to compute the inverse Laplace transforms:

Just like Laplace transforms have a linearity property, so do inverse Laplace transforms. “Given the two Laplace transforms F(s) and G(s), then L^{-1}\left\{aF(s)+bG(s)\right}=aL^{-1}\left\{F(s)\right\}+bL^{-1}\left\{G(s)\right\}.”

When trying to computer the inverse Laplace transforms, it is important to first look at the denominator and then try to identify the transform based on that. If you can’t figure it out just based on looking at the denominator, look at the numerator. Sometime you may have to manipulate the numerator to get into the correct form needed.


Let’s see an example (click here):




By looking at the table, we can see that the denominator is almost the same as the denominator of the transform for \sqrt{t}. With some algebraic manipulation to the numerator, we are able to figure out the inverse Laplace transform.

Here’s another example (click here):




See, Laplace transforms aren’t that hard after all. They can get a little messy and can be take a long time to solve after first, but with more practice, the better you’ll get. Make sure you have that table handy!

Until next time,
Leah

98 comments:

  1. how r u .... i am a mech. eng. and my son in faculty of eng.
    in fact my son asked me about how to proof laplace theory "laplace transform theory" academically .... i hope to get some help ... thank u very much for your effort
    [email protected]

    ReplyDelete
    Replies
    1. IEEE Project Domain management in software engineering is distinct from traditional project deveopment in that software projects have a unique lifecycle process that requires multiple rounds of testing, updating, and faculty feedback. A IEEE Domain project Final Year Projects for CSE system development life cycle is essentially a phased project model that defines the organizational constraints of a large-scale systems project. The methods used in a IEEE DOmain Project systems development life cycle strategy Project Centers in Chennai For CSE provide clearly defined phases of work to plan, design, test, deploy, and maintain information systems.


      This is enough for me. I want to write software that anyone can use, and virtually everyone who has an internet connected device with a screen can use apps written in JavaScript. JavaScript Training in Chennai JavaScript was used for little more than mouse hover animations and little calculations to make static websites feel more interactive. Let’s assume 90% of all websites using JavaScript use it in a trivial way. That still leaves 150 million substantial JavaScript Training in Chennai JavaScript applications.

      Delete
  2. With effective definitions and many other parts are almost giving more respective rewards hopefully this would indeed take you forward to the point which is generally considered to be so important. neurosurgery residency personal statement

    ReplyDelete
  3. You will find some info on how to buy essay online here. I recommend that you check it out as soon as possible

    ReplyDelete
  4. i was looking for a list of blogs for commenting thanks Ac market Gb instagram WPS Connect

    ReplyDelete
  5. I 'd create association that the sizable lion's share people mates unit endued with to unit getable at interims the least difficult place with loads of perfect the individuals who have extremely helpful things.
    Neet result
    Neet Login

    ReplyDelete
  6. NEET(UG) - 2018. As per regulations framed under the Indian Medical Council Act-1956 as amended in 2018 and the Dentists Act-1948 as amended in 2018.
    neet 2019 login
    neet result 2019

    https://neet365.in

    ReplyDelete
  7. This is really an amazing blog. Your blog is really good and your article has always good thank you for information.

    โปรโมชั่นGclub ของทางทีมงานตอนนี้แจกฟรีโบนัส 50%
    เพียงแค่คุณสมัคร Gclub กับทางทีมงานของเราเพียงเท่านั้น
    ร่วมมาเป็นส่วนหนึ่งกับเว็บไซต์คาสิโนออนไลน์ของเราได้เลยค่ะ
    สมัครสล็อตออนไลน์ >>> goldenslot
    สนใจร่วมลงทุนกับเรา สมัครเอเย่น Gclub คลิ๊กได้เลย

    ReplyDelete
  8. Thanks for sharing such a nice thing

    เว็บไซต์คาสิโนออนไลน์ที่ได้คุณภาพอับดับ 1 ของประเทศ
    เป็นเว็บไซต์การพนันออนไลน์ที่มีคนมา สมัคร Gclub Royal1688
    และยังมีหวยให้คุณได้เล่น สมัครหวยออนไลน์ ได้เลย
    สมัครสมาชิกที่นี่ >>> Gclub Royal1688
    ร่วมลงทุนสมัครเอเย่นคาสิโนกับทีมงานของเราได้เลย

    ReplyDelete
  9. Mathematics is interesting as well as difficult, it is great to get the advanced solutions, very helpful.
    Note to all: Protect your personal PCs & Laptops from data breaches caused by cyber attacks, use Norton antivirus program which you can get from norton.com/setup.

    ReplyDelete
  10. The Maharashtra Common Entrance Test popularly known as MHT-CET is a state level entrance exam organized by the Government of Maharashtra Common Entrance Test Cell. The exam is conducted for several Undergraduate courses in the subjects of Pharmacy, Engineering, Agriculture, Technology, and other Professional degree courses. The application process for the MHT-CET 2019 Exam tentatively will be closed on the final week of March 2019. The MHT-CET Examination 2019 will schedule to be held in the second week of May 2019. The MH-CET Exam will be held in 3 different stages i.e. Paper I, Paper II, and Paper III which will be conducted through pen and paper mode.

    ReplyDelete
  11. Needless to say, you may also cook your own recipes. For the price tag, in the event you only get a couple of good recipes it may be well worth it. If you are curious to know more about air fryer, visit here.

    ReplyDelete
  12. I really happy found this website eventually. Really informative and inoperative, Thanks for the post and effort! Please keep sharing more such blog.

    norton.com/setup

    norton.com/setup

    norton.com/setup

    office.com/setup

    roadrunner email

    mcafee.com/activate

    aol mail

    ReplyDelete
  13. Really great article, Glad to read the article. It is very informative for us. Thanks for posting.

    norton.com/setup
    mcafee.com/activate

    ReplyDelete
  14. To ease services,we avail web portal for pre written research papers services where US students make “write my essay” “write my research paper” and “write my paper” requests.

    ReplyDelete
  15. Are you facing any running error on Norton setup install Norton which scanning your windows or Mac, PC can contact our Norton tech engineer or visit Norton.com/setup. McAfee.com/activate

    Aol login
    Visit: www.mail.aol.com, aol login

    ReplyDelete
  16. Mcafee.com/activate – Get McAfee 25 digit Activation key to set up McAfee protection on your device. Visit www.mcafee.com/activate and follow the step-by-step guide to uninstall, download, install and activate McAfee Antivirus for your PC.
    Norton.com/setup - Buy the Norton product and then login to Norton account for downloading the setup. After installing it, navigate to www.norton.com/setup for activating it using the product key.
    office.com/setup ,Get Started. Setup & Install Microsoft Office Product Key from the official website. Visit www.office.com/setup and Follow the Instructions.

    ReplyDelete
  17. Well i am a regular reader of new blogs and i found this one is also such a nice one... keep it up
    mcafee.com/activate | mcafee.com/activate | norton.com/setup

    ReplyDelete
  18. You cannot imagine simply how much time I had spent for this information! Thanks!
    office.com/setup |
    norton.com/setup |
    mcafee.com/activate

    ReplyDelete
  19. Really great article, Glad to read the article. It is very informative for us. Thanks for posting.Norton™ provides industry-leading antivirus and security software
    for your PC, Mca, and mobile devices Visit @: - McAfee.com/activate | Norton.com/myaccount | norton.com/setup

    ReplyDelete
  20. norton.com/setup to install setup product key, Learn how to download, install, and put into outfill Norton Setup concerning speaking the subject of your Windows and Mac device.
    http://norton-comnortonnorton.com

    ReplyDelete
  21. Get all Steps for installation and download, activate MS office setup. Enter your Office product key on office.com/setup.
    http://office-comofficecom.com

    ReplyDelete
  22. After visiting office.com/setup login to microsoft account to take steps sticking together of covenant of office setup, Any complexity in office.com/setup our accomplished will calm for office install.
    http://officecomoffiice.com

    ReplyDelete
  23. Thanks for sharing your inspiring blog the information was very useful to my project ….

    mcafee.com/activate | norton.com/setup | norton.com/setup | office.com/setup | mcafee.com/activate | mcafee.com/activate

    ReplyDelete
  24. office 365 setup will help the users in activating, downloading and installing Microsoft Office on their PC . It will provide you the Step-by-Step guide for Microsoft office. If the subscription has expired, learn how to renew it. Get the complete guide to download, install and activate MS office setup.
    office.com/setup |
    Office.com/myaccount |
    Office 365 Setup

    ReplyDelete
  25. Office.com/setup present character staff are Officesetup-office-setup-show Certified Technicians dismissing don't everything Office setup considered hold any checks from any evacuated coming to if unequivocally showed up.
    http://officecom-officeoffice.com

    ReplyDelete
  26. You may on a freakishly central estimation get the Services in case you are a Service supporter on positive terms with an excellent 'ol bound, account with Office.com/setup.
    http://office-comofficeoffice.com/

    ReplyDelete
  27. Norton.com/setup is one among the basic standard antivirus that is tremendously looked for ensuring contraption and giving a 1 stop Norton setup security answer to any or each and every one of the total framework far and wide.
    http://nor-norton.com/setup/

    ReplyDelete
  28. McAfee Activate – Steps by steps for downloading, installing and activating your McAfee activate antivirus by redeeming the valid license keycode from – mcafee.com/activate
    Looking for norton.com/setup After you buy Norton Antivirus visit our site, sign in to norton account for Norton Setup or Install where you can enter and activate your 25 digit product key to setup your account – norton.com/setup. Get started by downloading and installing the setup. For activation, get the 25 characters alphanumeric product key from norton.com/setup. Norton offers hordes of reliable antivirus programs.
    Visit Office Setup, download the Microsoft Office file, and double-click the setup to install the product. Return to office.com/setup, sign in and enter the 25 character alphanumeric product key for activation

    ReplyDelete
  29. Microsoft Office setup consists of a setup file which the help of which a user can install their copy of Microsoft office.com/setup version on their computer and laptop. When you buy Office setup offline, then you have to install through a CD, and you get the setup file for installation if you get the office.com/setup online. If you use a Microsoft service like Outlook.com, OneDrive, Xbox Live, or Skype, you already have an office.com/setup. We have not been able to contact you for any product.

    ReplyDelete
  30. www.mcafee.com/activate Find easy steps for downloading, installing and activating McAfee, Enter your 25 digit activation code at www mcafee com activate.

    www.norton.com/setup - Simple steps to download and install Norton setup. Just enter www norton com setup Product key and get started with Norton.com/setup.

    ReplyDelete
  31. Thanks for the informative post that you have shared here.
    mcafee.com/activate

    ReplyDelete
  32. Download your Webroot SecureAnywhere product from here! Installation instructions available for all major platforms and operating systems.
    Webroot.com/safe

    ReplyDelete
  33. Avast login is an exceptionally notable security professional focus whose associations are accessible for a wide extent of gadgets including Windows PC, Mac, and Android Smart contraptions.
    Avast Login

    ReplyDelete
  34. MyDrive Connect is a free support application that helps you to manage your content and services on your TomTom navigation device.
    Tomtom Update

    ReplyDelete
  35. Garmin Express is computer software used to update maps and device software by downloading and installing the latest versions to your device.
    Garmin.com/express

    ReplyDelete
  36. Norton Utilities is a utility software suite designed to help analyze, configure, optimize and maintain a computer. The current version of Norton Utilities is Norton Utilities 16 for Windows XP/Vista/7/8.
    Norton.com/nu16

    ReplyDelete
  37. Facing problems with mcafee activate products? Get the best instant solution for purchasing, downloading, installing and activating McAfee key code.
    Mcafee.com/Activate

    ReplyDelete
  38. Learn how to purchase Norton product, set up a Norton account, download, install and activate Norton Setup with easy-to-follow steps. In order to activate your norton product key.
    Norton.com/setup

    ReplyDelete
  39. Norton designed and developed some other anti-virus by using their users such as sometimes perhaps Utility 16 when it comes to some utility program, that works on the device.
    norton.com/setup
    office.com/setup
    office.com/setup
    norton.com/setup
    norton.com/setup
    norton.com/setup

    ReplyDelete
  40. The geek squad support service is one of the primary reasons why I purchase tech products from Best Buy. Any time I've ever had to use my geek squad support Plans, I've had nothing but positive experiences."Geek Squad is the best! I love knowing my electronics are protected and there is no deductible for Trend Micro Geek Squad replacement or repair if I do have a problem.

    ReplyDelete
  41. I really happy found this website eventually. Really informative and inoperative, Thanks for the post and effort! Please keep sharing more such blog.

    norton.com/setup

    norton.com/setup

    mcafee.com/activate

    kaspersky activation code

    free pogo games

    roadrunner email

    aol mail

    ReplyDelete

  42. Office setup is a brand that needs no introduction whether you operate as a working professional in the corporate world, or a school student or operate through a remote workstation, there is no way for you to not have heard the name. Get your hands on office.com/setup by searching www.office.com/setup on your browser.
    www.office.com/setup
    office setup
    office setup
    www.office.com/setup

    ReplyDelete

  43. The Norton software is straightforward to install concerning speaking the colleague norton.com/setup. Among the spiteful characteristics of the Norton setup is that
    the system is automatically blocked by it.
    norton setup
    norton.com/setup.
    office setup
    www.office.com/setup

    ReplyDelete
  44. Hi, I’m Ben Charley. I’m a writer living in Newark, New Jersey. I am a fan of technology, music, and writing. I’m also interested in causes and painting.
    ATT toll free number

    ReplyDelete
  45. I am amazed by the way you have explained things in this article. This article is quite interesting and I am looking forward to reading more of your posts.
    Thanks for sharing this article with us.
    www.office.com/setup

    ReplyDelete
  46. For you currently it's been anything but difficult to arrangement Microsoft office in your framework. It wants each working framework and gadget including your cell phones. You can go to norton.com/setup to download the total office bundle and Enter office setup key code to enact office.
    Some Useful Links
    norton.com/setup|norton.com/setup|norton.com/setup|norton.com/setup|mcafee.com/activate

    ReplyDelete
  47. Whether you need a professional Android app or a user-friendly site for your business, hiring top web development companies in India is a good idea.

    ReplyDelete
  48. We are here providing services to resolve your problem of Printer. For any further update or help you can contact HP Support Number which is toll-free +1-888-845-6052.

    ReplyDelete
  49. I really happy found this website eventually. Really informative and inoperative, Thanks for the post and effort! Please keep sharing more such blog.

    norton.com/setup

    norton.com/setup

    mcafee.com/activate

    kaspersky activation code

    pogo not loading

    roadrunner email

    aol mail password recovery

    ReplyDelete
  50. It is free to register on our site and take great benefits of our complete range of services. Get latest information about If you are looking for Call Center jobs in mohali

    ReplyDelete
  51. We are here to help you in Bitdefender Removal Tool you can contact me anytime and for sure we will solve your mcafee antivirus problems.

    Call us: +1-888-845-6052




    Bitdefender Removal Tool

    ReplyDelete
  52. Your PC and laptop personal data is precious and we as webroot antivirus protection understand this very well. so

    we provide you the best antivirus cleaning program for better and secure protection from all threats. Please visit

    our site:

    Webroot.com/safe
    Office.com/setup

    ReplyDelete
  53. pes syringe filter

    PES syringe filters are widely applied in the biotechnological industries for the purification of enzymes and peptides that are easy to use and will serve all your laboratory filtration needs .

    ReplyDelete
  54. Thanks for sharing the blog it's very usefull. Hey I am Alice carter from New jersey. welcome to Geek Squad Technical Support. Geek Squad Support is There to Support You we have a Best Technician for Help you and solve your issues very quickly. Just Dial Geek squad tech support number 1-856-673-221. Our Technical Team available for 24/7 hours for support you.

    geek squad repair phone number
    Geek support number
    geek squad online support
    geek squad appointment cost
    geek squad repair prices
    geek squad repair phone number

    ReplyDelete
  55. Roku is connected to the Internet via a wireless or wired connection to your home network. Roku downloads videos from the Internet and then you watch videos on your TV. The video has not been saved due to its live streaming. You can also visit our website http://webslivesupport.com/roku-customer-service/

    AOL desktop gold download for windows 10
    AOL mail support phone number
    AOL customer service number
    roku customer service number
    adobe customer support number

    ReplyDelete
  56. Hello I really located you your blog is good and most helpful and useful www.mcafee.com activate download the post is good and very i visit your web page.it is useful and understand.

    ReplyDelete
  57. Thanks for sharing the blog. Hey I am Alicen Carter From New jersey. Welcome to Geek Squad Technical Support. Electronic devices such as Pcs, laptops, Tvs, are now an essential element of the modern workplace. Like another piece of equipment, electronic devices are prone to issues too. Geek Squad Support is There to Support You we have a Best Technician for Help you and solve your issues very quickly. Just Dial Geek squad tech support number 1-856-673-221. Our Technical Team available for 24/7 hours for support you.
    geek squad repair phone number
    Geek support number
    geek squad online support
    geek squad appointment cost

    ReplyDelete
  58. Assignment Help services offers quality and plagiarism-free work for students to acquire good grades on their subjects. If you are about to miss your due dates, login to greatassignmenthelp and complete your work on time.
    Assignment Help Online
    Online Assignment Help
    Assignment Help Online Services
    Assignment Helper
    Assignment Assistance
    Assignment Help Experts
    Online Assignment Help Services

    ReplyDelete
  59. Considering the mode from the move cantor, It truly is a great impressive clients. There's no spot pertaining to mediocrity. You need to indoctrinate most of your uprising buyers in order to reciprocate the clients. You cannot lull for your minimum amount, deleterious in order to touch your pleasures wouldn't it hauls for you whilst that happens in order to enlightenment matchless contingencys. trig calculator

    ReplyDelete
  60. When your website or blog goes live for the first time, it is exciting. That is until you realize no one but you and your. sydney splashbacks

    ReplyDelete
  61. Thanks for a very interesting blog. What else may I get that kind of info written in such a perfect approach? I’ve a undertaking that I am simply now operating on, and I have been at the look out for such info. killer promotional flyer

    ReplyDelete
  62. I wanted to thank you for this great read!! I definitely enjoying every little bit of it I have you bookmarked to check out new stuff you post. blog header in just one hour

    ReplyDelete
  63. Hi i read your blog is very interesting and helful.If you have any trouble if want more information norton.com/setup just contact us we will solve your problem.

    ReplyDelete
  64. We offer a wide variety of syringe filtersand pore sizes, as well as a selection of filter diameters to handle any sample volume.
    Simsii is specialize in Producing Syringe Filters and Syringe Filter

    ReplyDelete
  65. Simsii is specialize in Producing PES Syringe Filter for filteration needs.

    ReplyDelete
  66. This particular is usually apparently essential and moreover outstanding truth along with for sure fair-minded and moreover admittedly useful My business is looking to find in advance designed for this specific useful stuffs… 3D logo design

    ReplyDelete
  67. This is my first visit to your web journal! We are a group of volunteers and new activities in the same specialty. Website gave us helpful data to work. business logo

    ReplyDelete
  68. Acquiring Legit Research Paper Writing Services doesn’t have to drain your finances; College Essays for Sale charges are modest to ensure that we do not deter anyone from benefiting from our Purchase Research Paper.

    ReplyDelete
  69. Additionally, they will outline the qualities of Pay for Term Papers, and the advantages of hiring outstanding How to Write a Nursing Research Paper, as well as the qualities to consider when selecting this company for Legit Essay Writing Service.

    ReplyDelete
  70. Wow really good blog need more information.
    Hey I am los Bassett we provide you AOL technical support to guidance and helps for all kinds of doubts and queries that a person can have.
    AOL support number

    ReplyDelete
  71. office.com/setup - Here you can download the latest version of Office the link says 2016 but the version is most updated just visit www.office.com/setup. If you are facing issues then you can contact us at our office support number. we will instantly help you and solve issues.

    ReplyDelete
  72. I really happy found this website eventually. Really informative and inoperative, Thanks for the post and effort! Please keep sharing more such blog.

    norton.com/setup

    norton.com/setup

    mcafee.com/activate

    kaspersky activation code

    pogo not loading

    roadrunner email

    aol mail password recovery

    ReplyDelete
  73. Great post i must say and thanks for the information.I appreciate your post and look forward to more.

    norton.com/setup

    ReplyDelete
  74. best urologist doctor in chandigarh. Get extensive diagnosis and treatment for all urology problems like prostate problems, paediatric urology, reconstructive urology, uro oncolgy, female urology and many others by Kapoor's Kidney and Urostone hospital in Chandigarh, India with the help of their best urology specialist doctors.

    ReplyDelete
  75. ترجمه تخصصی کامپیوتر شاید از دید بسیاری از افراد کار بسیار دشواری باشد ولی متخصصان ما انرا برای شما به بهترین نحو برای شما انجام میدهند

    ReplyDelete
  76. Loaded with all these astounding highlights and functionalities, would prefer you not to download and introduce this Aol desktop gold Download . Basically pursue the underneath referenced strides to realize how to download and introduce or uninstall this product. Before that you should verify that your PC meets the framework necessities or not.

    ReplyDelete
  77. Students seek professional Custom Papers Writing assistance from Top Rated Term Papers Writingcompany to accomplish any type of essay assigned to them. When you request for urgent Professional Custom Research Paper, they strive to meet your professors’ requirements.

    ReplyDelete
  78. https://crackedlol.com/bitdefender-total-security-with-cracked-full-version/
    Bitdefender Total Security Crack can be a broad collection of tools. To guard your computer system against assorted kinds of dangers.

    ReplyDelete
  79. It was started as a dial-up service, but later expanded its business and became a major email service provider. Apart from allowing users to exchange emails, this AOL software has given its users many privileges. You need to visit our website http://webslivesupport.com/aol-desktop-gold/
    email support phone number
    email customer care numberr
    Download AOL desktop gold
    AOL desktop gold download

    ReplyDelete
  80. If you want go to the party you need a beautiful necklace, You can buy high quality necklace by visiting aixonne website, necklace for girlfriend birthday

    ReplyDelete