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Differential Equations, Separable, Simplification of answer

  1. Jun 29, 2015 #1
    1. The problem statement, all variables and given/known data
    I believe I have solved this differential equation, yet do not know how the book arrived at it's answer...

    Solve the differential equation in its explicit solution form.

    question.png

    The answer the book gives is...

    answer.png

    2. Relevant equations

    Separable Differential Equation

    3. The attempt at a solution

    dy/dx = x(x^2+1)/(4y^3)

    (4y^3)dy = (x^3+x)dx

    ∫(4y^3)dy = ∫(x^3+x)dx


    y^4 = 1/4x^4 + 1/2x^2 + c

    (initial condition, y(0) = -1/sqrt(2))

    (-1/sqrt(2))^(4) = 0 + 0 + c

    C = -1/4
    ....

    y^4 = 1/4x^4 + 1/2x^2 - 1/4

    y = (1/4x^4 + 1/2x^2 - 1/4)^(1/4)

    -----------

    I've experimented with simplifying this a bit and found a few other ways to express it, but nothing like what the book has written as the answer.
     
  2. jcsd
  3. Jun 29, 2015 #2

    RUber

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    Your sign on C is wrong.

    ##(\frac{x^4}{4}+\frac{x^2}{2} + \frac 14 ) = (\frac{x^2}{2} + \frac 12 )^2 ##
     
  4. Jun 29, 2015 #3

    SammyS

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    First. You made a mistake in finding C. What is (-1/sqrt(2))4 ? Fixing that will allow some factoring in the resulting expression.
     
  5. Jun 29, 2015 #4
    Oh wow, I don't think I would have seen that factor regardless. Thank you though. That helped a lot.
     
  6. Jun 29, 2015 #5
    Thank you, I understand now. In regards to the -, out front the answer from the book, I understand that comes from the square root, but how do they determine whether to go with the - or + solution. I haven't learned intervals of validity within the book yet...
     
  7. Jun 29, 2015 #6

    RUber

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    The radical implies the positive. Your initial condition forces the negative choice.
     
  8. Jul 4, 2015 #7
    Before rejecting an answer, you should plug it into the diff eq and see if it works.
     
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