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Problems with Laplace Transforms

  1. May 8, 2014 #1
    1. The problem statement, all variables and given/known data

    The coordinates ##(x,y)## of a particle moving along a plane curve at any time t, are given by

    [tex]\frac{dy}{dt} + 2x=\sin 2t,[/tex]
    [tex]\frac{dx}{dt} - 2y=\cos 2t.[/tex]

    If at ##t=0##, ##x=1## and ##y=0##, using Lapace transform show that the particle moves along the curve

    [tex]4x^2+4xy+5y^2=4[/tex]

    2. Relevant equations

    note: Lowercase letters ##x##,##y## are functions of ##t##. Uppercase letters ##X##,##Y## are functions of ##s##.


    3. The attempt at a solution

    Apply Laplace Transform on the given equations

    [tex]\frac{dy}{dt} + 2x=\sin 2t~~~~~~~~(1)[/tex]

    Applying LT,

    [tex]sY-y(0)+2X=\frac{2}{s^2+4}[/tex]

    [tex]2X+sY= \frac{2}{s^2+4}~~~~~~~~(2)[/tex]

    [tex]\frac{dx}{dt} - 2y=\cos 2t~~~~~~~~(3)[/tex]

    Applying LT,

    [tex]sX-2Y=\frac{s^2+s+4}{s^2+4}~~~~~~~~(4)[/tex]

    Now Solving ##(2)## and ##(4)## simultaneously,

    [tex]X=-s^3-s^2-4s-4[/tex]

    [tex]Y=2s^2+8[/tex]

    Now I have to apply Inverse Laplace transform to get back ##x## and ##y## but i dont know how to get ILT of a constant........... Also not sure everything I've done till now is right so please help
     
    Last edited by a moderator: May 8, 2014
  2. jcsd
  3. May 8, 2014 #2
    Sorry those fractions didn't come properly
     
  4. May 8, 2014 #3
    Solving Simultaneous Differential Equations using Laplace Transform

    1. The problem statement, all variables and given/known data

    The coordinates (x,y) of a particle moving along a plane curve at any time t, are given by

    [itex]\frac{dy}{dt}[/itex] + 2x=sin2t,
    [itex]\frac{dx}{dt}[/itex] - 2y=cos2t

    If at t=0, x=1 and y=0, using Lapace transform show that the particle moves along the curve

    4x2+4xy+5y2=4

    2. Relevant equations

    note: Lowercase letters x,y are functions of t. Uppercase letters X,Y are functions of s.


    3. The attempt at a solution

    Apply Laplace Transform on the given equations

    [itex]\frac{dy}{dt}[/itex] + 2x=sin2t -1

    Applying LT,

    sY-y(0)+2X=[itex]\frac{2}{s2+4}[/itex]

    2X+sY=[itex]\frac{2}{s2+4}[/itex] -2

    [itex]\frac{dx}{dt}[/itex] - 2y=cos2t -3

    Applying LT,

    sX-2Y=[itex]\frac{s2+s+4}{s2+4}[/itex] -4

    Now Solving 2 and 4 simultaneously,

    X=-s3-s2-4s-4

    Y=2s2+8

    Now I have to apply Inverse Laplace transform to get back x and y but i dont know how to get ILT of a constant........... Also not sure everything I've done till now is right so please help
     
  5. May 8, 2014 #4

    Borek

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    Staff: Mentor

  6. May 8, 2014 #5

    micromass

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    LaTeX fixed.
     
  7. May 8, 2014 #6

    Borek

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    Staff: Mentor

    Hmpf, I left that as an exercise to the reader.
     
  8. May 8, 2014 #7
    Your simultaneous solutions for X and Y are incorrect. Try again.

    Chet
     
  9. May 8, 2014 #8

    LCKurtz

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    Where did the -1 come from?
    Where did the -3 come from?

    If the -1 and -3 are supposed to be in there, you didn't transform them. And given the equations you have, how did you get those values for X and Y?
     
  10. May 9, 2014 #9
    those were eqn numbers.....eqn 1 and eqn 3.... I had left space but it didn't reflect in the post....I found my mistake.... it was in solving the simultaneous eqns.....you actually get X=(s+1)/(s^2+4)
    Y=-s/(s^2+4)
     
  11. May 9, 2014 #10
    thanx for your help
     
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