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: Nonexact Differential Equation (Possible to solve by integrating factor?)

  1. Apr 1, 2012 #1
    1. The problem statement, all variables and given/known data

    Solve the differential equation: [itex]t^2 y' + y^2 = 0[/itex]

    3. The attempt at a solution
    Now, it's definitely possible to solve this via separable of variables. But I am curious to know if I can solve it with an integrating factor. Having done some reading, I noticed that this equation is nearly in the form of an exact differential. Rewriting:

    [itex]t^2 y' + y^2 = 0 = t^2 \frac{dy}{dt} + y^2[/itex], implies,
    [itex]t^2 dy + y^2 dt = 0 = y^2 dt + t^2 dy[/itex].

    Unfortunately, letting [itex]M(x,y) = y^2[/itex] and [itex]N(x,y) = t^2[/itex] and then taking derivatives shows [itex]M_y = 2y ≠ N_t = 2t[/itex], so it looks like an exact equation isn't going to emerge from this.

    In the event that the equation is not exact, an integrating factor is typically sought. The problem is, I don't know how to go about finding such an thing. Can someone help me?
  2. jcsd
  3. Apr 1, 2012 #2
    You can try with this: [itex]\frac{dy}{y^2} + \frac{dt}{t^2} = 0[/itex] :wink:
  4. Apr 1, 2012 #3


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I think he know that.
    @TranscendArcu: It is not a given that a given first order DE can be solved by an integrating factor in any practical fashion, even if you can solve it by separation of variables.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook