1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Initial Value Problem using Laplace Transform help?

  1. Nov 1, 2015 #1
    1. The problem statement, all variables and given/known data
    Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teach is against using it..)

    y'' - 4y' + 3y = 0 ; y(0)=2 y'(0) = 8

    2. Relevant equations
    table.JPG
    Lf'' = ((s^2)*F) - s*f(0) - f'(0)
    Lf' = sF - f(0)
    Lf = F(s)

    3. The attempt at a solution
    My first attempt is of course realizing that the above equation is expressed in terms of 't'. So I must take the laplace transform on both sides.


    L(y'') - 4 L(y') + 3L(y) = L(0)

    [((s^2)*F) - s*f(0) - f'(0)] - 4 [ sF - f(0) ] + 3 [F(s)] = 0 (substituted with the above equations)

    (s^2)*F - 2s - 8 - 4s*F(s) + 8 + 3*F(s) = 0 (plugged in the initial values)

    Y(s)*[s^2 - 4s + 3] - 8 = 0

    Y(s) = 8 / [s^2 - 4s + 3]

    Knowing that s^2 + as + b = (s + a/2 ) ^ 2 + b - ((a^2)/4)
    I get: (s-2)^2 - 1 for the denominator.

    Y(s) = 8 / [(s-2)^2 - 1 ]


    At this final step....I am not sure how I will be able to transform to Y(t) again if I am not allowed to use "Partial Fractions"? Any assistance would be truly appreciated. Thanks!



    The solution for this problem is :
    y = 2(e^2t)*cosht + 4(e^2t)*sinht
     
    Last edited: Nov 1, 2015
  2. jcsd
  3. Nov 1, 2015 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    I would immediately take off marks because you do not stick with one single notation--either y or F or f, but not y as well as both F and f and in the same problem and standing for the same quantity.

    Anyway, you can get the inverse transform of 1/(s^2 - 1) from your table; and you ought to know the relationship between the inverse transforms of G(s) and G(s-2). You really do not need partial fractions, and the instructor is right in forcing you to avoid them---it makes you learn to use some important properties of LTs that will be useful to you later in life.
     
  4. Nov 1, 2015 #3
    Using the shift theorem from Y(s) = 8 / [(s-2)^2 - 1] , I get Y(t) = 8 * L^(-1){1 / (s^2 - 1)}.... 8*sinht*e^t.

    I still don't know how my teacher got y = 2(e^2t)*cosht + 4(e^2t)*sinht
     
  5. Nov 1, 2015 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Well, for one thing, you have an incorrect ##Y(s)##: the denominator is OK but the numerator is wrong.
     
  6. Nov 1, 2015 #5
    My mistake.
    Never mind, I solved it. Thank you!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Initial Value Problem using Laplace Transform help?
Loading...