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Differential Equations, Separable, Explicit Solution

  1. Jun 29, 2015 #1
    1. The problem statement, all variables and given/known data

    Solve the differential equation, explicitly.

    dy/dx = (2x)/(1+2y)

    The answer given by the book is...

    -1/2 + 1/2sqrt(2x - 2x^2 +4)

    2. Relevant equations

    Process for solving separable differential equations

    3. The attempt at a solution

    dy/dx = (2x)/(1+2y)

    (1 + 2y)*dy = 2x*dx

    ∫(1+2y)*dy = ∫2x*dx

    y + y^2 = x^2 + C

    ....

    I seem to have solved it, implicitly, but I see no way of solving it explicitly.
     
  2. jcsd
  3. Jun 29, 2015 #2

    Mark44

    Staff: Mentor

    Use the quadratic formula to solve for y in terms of x. Note that the quadratic formula will give you two solutions. Your book seems to have chosen one of the solutions arbitrarily.
     
  4. Jun 29, 2015 #3
    Ughhh, I thought about that, I just never applied it... Thank you so much for your help!

    P.S. Should I use the C(constant) as the C in the quadratic formula? or just leave it on the other side with the x?
     
  5. Jun 29, 2015 #4

    Mark44

    Staff: Mentor

    Is there an initial condition in your problem that you didn't include? The book's solution doesn't include the constant, which makes me suspect that they are using information not shown here.
     
  6. Jun 29, 2015 #5
    Yes, I'm sorry I forgot to include the initial condition, y(2) = 0

    P.S. Never mind, I figured out how to do the quadratic with this equation!
     
  7. Jun 29, 2015 #6

    Mark44

    Staff: Mentor

    Also, are you sure you wrote the problem down correctly? I don't get the same solution as you showed.
     
  8. Jun 29, 2015 #7
    This is the solution I ended up attempting...


    dy/dx = (2x)/(1+2y)

    (1 + 2y)*dy = 2x*dx

    ∫(1+2y)*dy = ∫2x*dx

    y + y^2 = x^2 + C

    (initial condition) 0 + 0 = 2^2 + C, C = -4

    y^2 + y + (4 - x^2) = 0

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

    y = (-1 +/- sqrt(x^2 - 15))/2

    y = -(1/2) +/- (1/2)sqrt(x^2-15)


    The only problem I'm having is, how did they know to only use the one solution, instead of both of them, we didn't learn about intervals of validity or anything yet.
     
    Last edited: Jun 29, 2015
  9. Jun 29, 2015 #8

    Mark44

    Staff: Mentor

    You have what I got, which is different from the solution you posted. It's possible they have a typo in their answer, or even that it is just flat wrong. You can verify that your answer is correct by differentiating it to show the dy/dx = (2x)/(1 + 2y) is identically true. I checked the book's answer and it didn't satisfy the differential equation.
     
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