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Separate Variables Differential Eq. of Cubic Power

  1. Jun 4, 2013 #1
    1. The problem statement, all variables and given/known data
    When possible express the general solution in explicit form.
    Solve dy/dx =x^2 /(1+y^2)

    2. Relevant equations
    This is a first order non-linear ordinary differential equation.


    3. The attempt at a solution
    dy(1+y^2) = x^2 dx
    Integration both sides returns:
    y+ (y^3 )/3= (x^3)/3 +C
    Now, I am aware that there is more than one solution for y involving imaginary numbers. Can someone help me in the next step or direct me to a site?
    I have seen cubic solutions tutorial, but they involve equations of the form: Ax^3+Bx^2+Cx+D=0

    Thank you.
     
  2. jcsd
  3. Jun 4, 2013 #2

    Dick

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    Science Advisor
    Homework Helper

    I really don't think you want to solve for y. With an expression like that I'd just leave in the implicit form you already have, or maybe express x as a function of y instead. Don't try to use the cubic formula. It's a mess.
     
    Last edited: Jun 4, 2013
  4. Jun 4, 2013 #3
    Yes, I am aware that it is kinda difficult to solve for 'y' and that's why I wanted to try it out. It involves imaginary numbers and many roots.
    If someone can point me in the right direction, that would great.
     
  5. Jun 4, 2013 #4
    Luckily, this is a seperable equation, which means you can rewrite it as
    $$(1+y^2)dy = x^2dx.$$
    Now, what can you do to get rid of those pesky differentials?
     
  6. Jun 4, 2013 #5

    tiny-tim

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  7. Jun 4, 2013 #6
    Thank you, tiny-tim. I usually use latex for big equations, but I thought it wouldn't be a big deal.

    I was thinking that I could solve it like your wikipedia link... this will be interesting. Thanks.
     
  8. Jun 4, 2013 #7

    Mark44

    Staff: Mentor

    You really need to read through the thread. The OP has already done this and has gotten a solution.
     
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