1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Find solution of initial value problem - 1st order non-linear ODE

  1. Jun 23, 2011 #1
    we have to solve the following problem for our ODE class.

    1. The problem statement, all variables and given/known data

    Find the solution of the initial value problem
    dx/dt = (x^2 + t*x - t^2)/t^2

    with t≠0 , x(t_0) = x_0

    Describe the (maximal) domain of definition of the solution.

    3. The attempt at a solution
    Well, I know that this is a 1st order nonlinear ODE. Unfortunately I got no clue how to deal them.
    I tried this:
    dx/dt = (x^2 + t*x - t^2)/t^2
    = x^2/t^2 + x/t -1

    Now substitute: u = x/t -> x=ut , x'=u't+u
    Therefore we get:
    u't+u = u^2+u-1
    t* du/dt +u = u^2+u-1 //-u
    t* du/dt = u^2 -1

    0= t*u' -u^2 +1
    which is my dead end.

    Is the idea ok? What could I do?

    Kind regards,

    PS: How can i insert a fraction?
  2. jcsd
  3. Jun 23, 2011 #2
    I just noticed that we might be able to solve this via applying the Riccati equation????
  4. Jun 23, 2011 #3


    User Avatar
    Homework Helper

    This is a separable differential equation, you must be able to solve it
    :smile: !

  5. Jun 23, 2011 #4
    As far as I know it is a separation of variables case, iff there would be no t in front of the du/dt. Or no -1.
    You think of the solution
    du / (u^2 -1) = dt / t ,
    don't you?

    But anyway thanks for your answer =)
  6. Jun 23, 2011 #5


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    That doesn't matter. Just keep the t with the dt.
    That's exactly what he is thinking of. Use partial fractions on the left side and integrate both sides.
  7. Jun 24, 2011 #6


    User Avatar
    Homework Helper

    I do. And you should be able to integrate both sides.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook