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

A couple questions

  1. Dec 15, 2013 #1
    1. For a non-homogeneous, linear, 2nd order DE, using the method of undermined coefficients, what do you use for the particular solution for [itex]e^x[/itex] and [itex]cosx[/itex] or [itex]sinx[/itex]. For example, if it is [itex]x^2[/itex], you use [itex]Ax^2+Bx+C[/itex].

    2.For [itex]y''-2y'+2y=0[/itex] i put [itex]y(x)=c_1e^{(1+i)x}+c_2e^{(1-i)x}[/itex] but got it wrong. The right answer was written as [itex]y(x)=e^{ix}(c_1cosx+c_2sinx)[/itex]. I always assumed that an answer using [itex]e^x[/itex] was right. How do I know when to use [itex]e^x[/itex] and when to use [itex]sinx[/itex] or [itex]cosx[/itex]?

    Thanks a bunch!!
  2. jcsd
  3. Dec 15, 2013 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper

    You use the solution that fits with the rest of the DE.

    How do you know in advance? By experience. Painful experience. Builds character.
    That's why you are given these exercises - to gain the experience.

    Have you seen:

    i.e. if the inhomogeniety is e^x, then try Ae^x as a particular solution.

    For y''-2y'+2y = 0, you guess ##y=e^{\lambda x}## - which ends up with your solution right?
    - expand the "right" answer in terms of powers of e and see if it matches yours.
    Last edited: Dec 15, 2013
  4. Dec 15, 2013 #3
    Thanks Simon. I'm still a little confused. What do you mean by expanding it in terms of powers of e?
  5. Dec 15, 2013 #4

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper

    You can express sine and cosine as a sum of complex exponentials.
    Look up Euler's formula.

    general solutions of the form ##y=Ae^{\lambda_1x}+Be^{\lambda_2 x}##
    and of form ##y=e^{ax}C\sin(x)+D\cos(x)## should be equivalent in this case.
  6. Dec 15, 2013 #5
    Oh ok i see. Thanks
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook