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

Why is general solution of homogeneous equation linear

  1. Oct 30, 2011 #1
    Hi, I don't understand why the general solution of 2nd order homogeneous equation is linear? Why is c_1e^(xt)+c_2e^(xt) a linear differential equation? What am I missing here? Any help would be appreciated, I'm struggling a bit understanding the concepts of differential equations...
     
  2. jcsd
  3. Oct 31, 2011 #2
    The general solution is a linear combination of two linearly independent solutions y1(x) and y2(x).


    p.s. I think something is not right with your notation e^(xt) .
     
  4. Oct 31, 2011 #3
    Yes, you're right. The x in the exponent of e should be r, where you would find the roots using the characteristic equation. C_1e^(r_1t)+C_2e^(r_2t). So this is a linear solution because y_1 and y_2 are to the first power? Even though the function e^rt is not a linear function?
     
  5. Oct 31, 2011 #4

    lurflurf

    User Avatar
    Homework Helper

    It is linear in y not x.
    L is a linear operator if
    L[Ʃanyn]=ƩanL[yn]
     
  6. Oct 31, 2011 #5
    Ah, OK. Thanks guys.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook