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Math Help - Laplace Transform Questionn

  1. #1
    Super Member
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    Laplace Transform Questionn

    Hi

    I got a couple of questions that i am having trouble with:

    1)Use Laplace transform to solve the following initial value problems:

    y''-2y'+2y=c

    y(0) = 20 and y'(0) = 34

    This is what i have done:

    (s^2 Y(s) - sy(0) - y'(0)) - 2(sY(s)-y(0))+2Y(s) = X = \frac{2-cos(t)}{2s}

    (s^2 Y(s) -20s - 34) - 2(sY(s)-20)+2Y(s) = \frac{2}{(s-1)}+\frac{6}{(s-1)^3}

    (s^2 - 2s + 2 )Y -20s - 34 +40 = \frac{2}{(s-1)}+\frac{6}{(s-1)^3}

    (s^2 - 2s + 2 )Y = \frac{2}{(s-1)}+\frac{6}{(s-1)^3} + 20s - 6

    (s^2 - 2s + 2 )Y = \frac{2(s-1)^2}{(s-1)}+\frac{6}{(s-1)^3} + 20s - 6

    (s^2 - 2s + 2 )Y = \frac{6+ 2(s-1)^2}{(s-1)^3} + 20s - 6

    Y = \frac{8+2s^2-4s}{(s-1)^3 (s^2 - 2s + 2 )} + \frac{20s - 6}{(s^2 - 2s + 2 )}

    What should i do next??

    2) Use Laplace transform to solve each system of differential equations
    \frac{dx}{dt} = -y+cos(t)

    \frac{dy}{dt} = x+1

    given x(0) = 1 y(0) = 0

    This is what i have done

    sX - 1 = -Y + cos(t)
    sY = X + 1

    sX + Y = 1+ cos(t)
    sY - X = 1

    Matrix Form to solve X and Y

    | s 1 |
    | s -1 |

    Det= -2s


    | 1+cos(t) 1 |
    | 1 -1 |

    Det_x = -2-cos(t)

    | s 1+cos(t) |
    | s 1 |

    Det_y = -scos(t)

    X = \frac{2-cos(t)}{2s}
    Y = \frac{cos(t)}{2}
    what is wrong?

    P.S
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Paymemoney View Post


    2) Use Laplace transform to solve each system of differential equations
    \frac{dx}{dt} = -y+cos(t)

    \frac{dy}{dt} = x+1

    given x(0) = 1 y(0) = 0

    This is what i have done

    sX - 1 = -Y + cos(t)
    sY = X + 1

    sX + Y = 1+ cos(t)
    sY - X = 1

    Matrix Form to solve X and Y

    | s 1 |
    | s -1 |

    Det= -2s


    | 1+cos(t) 1 |
    | 1 -1 |

    Det_x = -2-cos(t)

    | s 1+cos(t) |
    | s 1 |

    Det_y = -scos(t)

    X = \frac{2-cos(t)}{2s}
    Y = \frac{cos(t)}{2}
    what is wrong?

    P.S
    You have not taken the LT of \cos(t)

    CB
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