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Math Help - few questions with laplace transform

  1. #1
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    few questions with laplace transform

    I am having trouble this this problem.
    y' + 6y = e^4^t,  y(0) = 2

    Heres the work I have up until where I get stuck:
    SY(s) - y(0) + 6Y(s) = \frac{1}{s-4}
    SY(s) - 2 + 6Y(s) = \frac{1}{s-4}
    SY(s) + 6Y(s) = \frac{1}{s-4} + 2
    Y(s)(s+6) = \frac{1}{s-4} + 2
    Y(s) = \frac{1}{(s-4)(s+6)} + \frac{2}{s+6}
    partial fractions:
    \frac{1}{(s-4)(s+6)} + \frac{2}{s+6} = \frac{A}{s-4} + \frac{B}{s+6}

    How do I solve this partial fraction equation now? I have tried it and I am ending up with a different answer than the solutions manual for B.

    My other question is:
    Find the fundeamental set of solutions using only real coefficients: y^(^4^) + 4y = 0

    I get the steps:
    m^4 + 4 = 0
    m = \sqrt[4]{-4}
    y1 = e^xcosx
    y2 = e^xsinx
    y3 = e^-^xcosx
    y4 = e^-^xsinx

    I am lost as to where you get m = \sqrt[4]{-4} is equal to \pm 1 \pm i and how this works on a graph....

    Thanks for the help guys, I really appreciate it!
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  2. #2
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    You cause me pain. How can you even SAY "laplace transform" and still be struggling with this basic algebra?

    Common Denominator. Add. Equate like terms. {shakes head}

    I get A = 1/10 and B = 19/10. What do you get and how?
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  3. #3
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    Quote Originally Posted by Jeonsah View Post
    My other question is:
    Find the fundeamental set of solutions using only real coefficients: y^(^4^) + 4y = 0

    I get the steps:
    m^4 + 4 = 0
    m = \sqrt[4]{-4}
    y1 = e^xcosx
    y2 = e^xsinx
    y3 = e^-^xcosx
    y4 = e^-^xsinx

    I am lost as to where you get m = \sqrt[4]{-4} is equal to \pm 1 \pm i and how this works on a graph....

    Thanks for the help guys, I really appreciate it!
    Your four solutions for m are wrong.

    m^4 + 4 = 0

    m = (-4)^{1/4} = \sqrt{2} \cdot (-1)^{1/4}

    Recall that -1 = e^{i \pi} = e^{i \pi + i2n \pi}

    So
    m = (-4)^{1/4} = \sqrt{2} \cdot (e^{i \pi + i2n \pi})^{1/4} = \sqrt{2} \cdot e^{i \pi / 4 + in \pi /2}

    m = \sqrt{2} ( cos( \pi / 4 + n \pi / 2) + i~sin( \pi / 4 + n \pi / 2) )

    So the four solutions to m are, using n = 0, 1, 2, and 3:
    \{ cos(\pi / 4) + i~sin(\pi / 4),~cos(3 \pi / 4) + i~sin(3\pi / 4),~cos(5 \pi / 4) + i~sin(5\pi / 4),~cos(7 \pi / 4) + i~sin(7\pi / 4) \}

    I'll let you make the solutions look nicer.

    -Dan

    Edit: Whoops! I forgot about the factor of sqrt(2). Okay, so the final (simplified) solutions for m will be 1 + i, -1 + i, -1 - i, and 1 - i. Notice that this does generate the solution series given by your professor.
    Last edited by topsquark; May 7th 2011 at 08:21 PM.
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  4. #4
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    The reason why i was lost with the partial fraction part was because of the  \frac{2}{s+6} Exactly what do you do with it? do you combine it with the term on the left of the equal side, or do you move it over?
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Jeonsah View Post
    The reason why i was lost with the partial fraction part was because of the  \frac{2}{s+6} Exactly what do you do with it? do you combine it with the term on the left of the equal side, or do you move it over?
    You get a common denominator and you add it.
    \frac{1}{(s - 4)(s + 6)} + \frac{2}{s + 6} = \frac{1}{(s - 4)(s + 6)} + \frac{2(s - 4)}{(s - 4)(s + 6)}

    Finish adding the fractions, then use partial fractions on it.

    -Dan
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  6. #6
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    ah ok, thats where I got tripped up. I was adding it wrong. Thanks for helping me solve the second problem. That makes a lot more sense than how my teacher did it. Gracias Senor!
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  7. #7
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    TKHunny, sorry for being a noob. Sometimes you just forget somethings you know? Ive been studying so much that im getting delirious. But none the less {shakes head} = me slapping myself!
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  8. #8
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    I hope you don't mind my friendly ribbing. If I received a paper like this during a class, I would mark it, "I should fail you for the entire semester for this".

    One must remember ones algebra! :-)
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  9. #9
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    Quote Originally Posted by TKHunny View Post
    I hope you don't mind my friendly ribbing. If I received a paper like this during a class, I would mark it, "I should fail you for the entire semester for this".

    One must remember ones algebra! :-)
    And if I was a head of a department, I would write a note saying "I should fire you for writing such a demotivational statement on a student's work..."
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  10. #10
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    Quote Originally Posted by Prove It View Post
    And if I was a head of a department, I would write a note saying "I should fire you for writing such a demotivational statement on a student's work..."
    Ah, but if you were in my class, or if you were my department head, you would never take it as demotivational. You just can't see the huge smile on my face, the care and compassion in my eyes, and the personal commitment to make sure each student who cares is allowed to succeed and enabled to do so.
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  11. #11
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    Quote Originally Posted by TKHunny View Post
    Ah, but if you were in my class, or if you were my department head, you would never take it as demotivational. You just can't see the huge smile on my face, the care and compassion in my eyes, and the personal commitment to make sure each student who cares is allowed to succeed and enabled to do so.
    That is true, but the department head or principal wouldn't be able to see that either - any complaints and the only tangible evidence is what's written on the paper...
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