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How to integrate this equation?

  1. Oct 17, 2011 #1
    Dear all,

    Someone could help me to understand how I can resolve the following equation :

    dV/dt= A V^2 + B V + C

    Where V :V(t), A(t), B(t), C(t)

    Is there any method or indications about this ?

    Thanks in advance,

  2. jcsd
  3. Oct 17, 2011 #2


    Staff: Mentor

    Separate the variables.

    Rewrite the equation as
    dV/(A V^2 + B V + C) = dt, and then integrate.

    Is this a homework problem?
  4. Oct 17, 2011 #3


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    I don't think that will work if the A, B, and C are functions of t as the OP indicates. If C(t) is identically 0 it might be solvable as a Bernoulli equation. In general there's no hope for an explicit solution.
  5. Oct 17, 2011 #4


    Staff: Mentor

    Sorry, I totally missed it that A, B, and C were functions of t.
  6. Oct 18, 2011 #5
    Thanks a lot Mark44 and LCKurtz.

    C is also function of time. In that case, if there is not hope for explicit solution, should I try to solve it numerically? and if yes? which method could I use?

    I know how varies A, B and C as a function of another variable "Z". In other words:

    dV/dt= A(Z) V^2 + B(Z) V + C(Z)


  7. Oct 18, 2011 #6
    You are basically solving it for V as a differential equation, is that correct? Wolfram Alpha calls it a Riccati equation. The answer looks eerily similar to the quadratic formula, with an arctan involved, so that might be a hint as to how to go about solving a portion of it.
  8. Oct 18, 2011 #7

    D H

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    Sometimes you have to be very explicit with Mathematica. This is one of those times. You should have told Mathematica that z is a function of t: http://www.wolframalpha.com/input/?...V/dt=+a(z(t))*V^2+++b(z(t))*V+++c(z(t))+for+V

    Yes, this is a Riccati equation, and no, there isn't a general closed-form solution. There are special cases for which a closed-form solution is known.
  9. Oct 18, 2011 #8
    I thought he had said z here was a independent variable, and not a function of t? (edit) Nevermind... Different versions of the same formula floating around, I looked at the last one.
  10. Oct 19, 2011 #9
    dV/dt= A(Z) V^2 + B(Z) V + C(Z)

    if Z is a function of t there is no known analytical method to solve it (except in some particular forms of functions A(Z(t)), B(Z(t)), C(Z(t)). You have to solve it numerically.

    if Z is not function of t, then it can be analyticaly solved :
    t = integral of dV/(A V² + B V + C)
    For a given value of Z which dosn't depends on t, A(Z), B(Z), C(Z) are constants.
    So you obtain t as a function of V and Z
    Inverting this fonction would lead to V as a function of t and Z
    But if, afterwards, you make Z varying with t, the result would be false.
  11. Oct 20, 2011 #10
    Dear All,

    Thanks a lot to everybody for the advices. I learn a lot in this forum always...

    Yes! I see that this is a Ricatti equation and I think the best is to solve it with numerical methods... asssuming the initial conditions for each one the variables depending on "t".

    Thanks again,

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