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Thread: Help with using integrating factor to solve dy/dt = 2x + 4t (ans: -2t -1 + ce^(2t))

  1. #1

    Help with using integrating factor to solve dy/dt = 2x + 4t (ans: -2t -1 + ce^(2t))

    Hi,

    First post here, so bear with me if I'm breaking forum etiquette

    I'm working out exercises in my Croft & Davis engineering mathematics book. I'm supposed to solve the equation :

    dy/dt = 2x + 4t

    By using an integrating factor. According to the book, the answer should be :

    -2t -1 + ce2t

    While my answer is just -2t + ce2t. What am I doing wrong? And where does the -1 come from?

    Thank you!

  2. #2
    Elite Member stapel's Avatar
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    Cool

    Quote Originally Posted by MisterMD View Post
    I'm supposed to solve the equation :

    dy/dt = 2x + 4t

    By using an integrating factor. According to the book, the answer should be :

    -2t -1 + ce2t

    While my answer is just -2t + ce2t. What am I doing wrong? And where does the -1 come from?
    Unfortunately, it is not possible to troubleshoot work that we cannot see. So please reply with a clear listing of all of your steps. Thank you!

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    Elite Member
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    Quote Originally Posted by MisterMD View Post
    Hi,

    First post here, so bear with me if I'm breaking forum etiquette

    I'm working out exercises in my Croft & Davis engineering mathematics book. I'm supposed to solve the equation :

    dy/dt = 2x + 4t
    Is this the correct ODE? You are showing 3 variables!
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  4. #4
    Ah, sorry for not replying. I actually figured it out though! Again, I hope to make it clear here. Bear with me. Also thanks to Wolfram Alpha.

    It's solving the equation by using an integrating factor. The first thing I did was then make it into the standard form dy/dx + P(x) = Q(x). So I turned dx/dt = 2x + 4t into dx/dt - 2x = 4t.

    P is then -2, Q is 4t.

    Then I worked out the integrating factor (called U) : e(integral(P) dx) which worked out to e-2t.

    Multiplying everything with this factor makes it into this : (dx/dt * e-2t) - (2xe-2t) = (4te-2t). Using the product rule, that can be simplified to :

    d(xe-2t)/dt = 4te-2t

    Integrate both to produce : xe-2t = integral(4te-2t)dt, which is rather hairy. I did it with integration by parts.

    4 (integral(te-2t)dt. Removing the factor of 4. Integration by parts is : Integral(UV') = UV - Integral(U'V)

    U = t, U' = 1
    V = (-e-2t)/2, V' = e-2t

    So it becomes : (-te-2t)/2 - Integral((-e-2t)/2)

    Becoming : (-te-2t)/2 - (e-2t)/4 + C


    Rmoving fractions (multiplying all by 4) : -2te-2t - e-2t + C

    And then the final step which is dividing all by U : x = -2t -1 + Ce2t which is exactly the solution in the book. Still! Thanks for replying

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