Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

(probably stupid) Differential equation question

  1. Feb 21, 2009 #1
    Hey all, i just lost a TON of points on a test for solving a

    problem in a way that is apparently invalid.

    The problem was verify that y(x) = x+1 is a solution for dy/dx =

    y*y-x*x-2x; y(0) = 1. i plugged y = x+1 into the right side of

    the second equation, got dy/dx = 1, integrated to get y=x+c, used

    y(0) = 1 to get c= 1, therefore y = x + 1

    My professor's annoyed 2-second explanation about why my method

    is invalid was that I assumed that it worked to prove that it

    worked. I sorta buy it, but I'm not completely convinced, could

    someone give me a counter example to prove that my method is not

    legit? [to clarify, my method is to plug in y(x) into the DE,

    then integrate, then use the given initial conditions to solve for

    c to get a new y(x) and make sure that my new y(x) is the same as

    the old one].

    The counter example I am requesting would take a form that is

    similar to the problem above, except that y(x) would not be a

    legit solution to dy/dx, BUT my method would falsely show that

    y(x) does work. Obviously, if no such counter example exists,

    that my method proves that the DE works and I should not have lost

    any points

    Thanks in advance!
  2. jcsd
  3. Feb 22, 2009 #2


    Staff: Mentor

    What you did was sort of a cross between verifying that a given function was a solution and attempting to find the solution.

    When you substituted y = x + 1 into the right side to get 1, why didn't do the same substitution on the left side? After all, if y = x + 1, dy/dx = 1.

    When you got dy/dx = 1, that's not the same differential equation as the one you started with. dy/dx happens to be equal to 1 when y = x + 1. By treating dy/dx as a constant, you are eliminating all of the other potential soltutions of the DE dy/dx = y^2 - x^ - 2x. Off the top of my head I don't know what the other solutions to this DE might be, but such an equation (without the initial condition) generally has an infinite number of them.

    There's a big difference between being asked to verify that a function is a solution of an initial value problem (a DE + a set of initial conditions), and finding the solutions to a DE. At this point, you probably don't have the tools to solve nonlinear DEs like this one, so make your life a little easier and do what is asked for. After you've done that, you can explore alternate techniques.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook