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First order ode question

  1. Jun 23, 2006 #1
    I would like to solve a problem of the type

    (da/dt)^2 + f(a)* (da/dt) = g(a) (1)

    a=a(t) unknown function
    f(a), g(a) = known functions of a.

    This differential equation is a first order ODE but (da/dt)^2 makes it different compared to a typical first order ODEs (at least to my knowledge)

    I would like to find a(t) satisfying (1) subject to certain initial conditions (say a(0.1)=2).

    I feel that no appropriate analytical solution exists for this type of problem, so I am looking for a numerical method to integrate it.

    I am thinking of setting da/dt=y thus having
    ---------------------------
    y^2 + f(a)* y = g(a)
    da/dt=y
    ----------------------------

    and then writing da/dt = ( a(i+1) - a(i) ) /dt

    so the problem becomes
    ---------------------------
    y^2 + f(a(i))* y = g(a(i)) (2)
    a(i+1) = dt*y + a(i) (3)
    ----------------------------

    Now I am thinking of solving (2) for the value of y which corresponds at i=0 and then keep one the two solutions (which one to keep is not very clear ….(or if they are both imaginaray?)) Then with a selected small dt (say dt=0.001) find a(i+1). Then continue the iteration scheme this way.

    I know a priori that da/dt is positive and thus a(t) is an increasing function of t.

    I would like to have opinion from you whether the previous reasoning is TOTALLY WRONG or not. If it wrong I would appreciate if you just give a hint of how to attack the problem

    Thanks
     
  2. jcsd
  3. Jun 24, 2006 #2

    Tide

    User Avatar
    Science Advisor
    Homework Helper

    HINT: Use the quadratic formula. :)
     
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