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Differential Eq. Last Step Solution Separating Variables

  1. Jun 3, 2013 #1
    Solve each differential equation. Express the general solution in explicit form.

    y' = (3x^2 -1) / (3+2y)

    So, I will skip many steps, because they are easy. However, I am stuck in one of the last ones.
    y^2 +3y = x^3- x +C

    y(y+3)= x^3 - x +C

    I have seen the solution for y, but I don't understand how it is derived. Can someone help?
  2. jcsd
  3. Jun 3, 2013 #2
    Try completing the square on the left hand side, and then taking the square root of both sides of the equation.
  4. Jun 5, 2013 #3


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    y(y+ 3)= y^2+ 3y= x^3- x+ C

    y^2+ 3y+ (x- C)= 0.

    Solve that quadratic equation using the quadratic formula.
  5. Jun 5, 2013 #4
    Hello HallsofIvy, How did you get rid of x^3 ?
  6. Jun 5, 2013 #5
    He appears to inadvertently left it out. The method HallsofIvy suggested and the method I suggested are mathematically equivalent.

  7. Jun 6, 2013 #6


    Staff: Mentor

    Move all of the terms to the left side, and you'll have a quadratic in y. It's probably simpler at this point to just use the Quadratic Formula to solve for y, which will have two parts separated by ±, as is usually the case.

    If there is an initial condition, it might lead you to choose one or the other of the two values.
  8. Jun 6, 2013 #7


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    I waved my magic wand and uttered a spell of "Carelessness"!
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