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

How does the Wronskian determine uniqueness of solution?

  1. Oct 26, 2014 #1
    Given an nth order DE, how (intuitively and/or mathematically) does computing the Wronskian to be nonzero for at least one point in the defined interval for the solution to the DE ensure the solution is unique and also a fundamental set of solutions?

    Also, is it true that if W = 0, it is 0 for all entries, and also if W does not equal 0, it is never equal to 0? If so, is there a formal proof (and/or intuitive explanation) for this?

    Also, for a non-homogeneous DE, the solution is in the form Yhomog + Ypart = y, where Ypart is a solution to the g(t). Does Ypart necessarily have to have ALL solutions or just one?
     
    Last edited: Oct 26, 2014
  2. jcsd
  3. Oct 26, 2014 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You want to look at Abel's identity. Look here: http://archive.lib.msu.edu/crcmath/math/math/a/a009.htm
    Basically it shows the Wronskian is a constant times an exponential type function, so it's always zero or never zero.

    Call your DE ##L(y) = f(x)##. If it is a second order DE then ##y_c = Cy_1 + Dy_2## is the general solution of the homogeneous equation. Suppose ##y_p## is a particular solution of the NH equation ##L(y_p) =f(x)##. Then if ##y_q## is any solution of the NH equation you have ##L(y_q)=f(x)## so ##L(y_q-y_p) = f(x) - f(x) = 0##. This says ##y_q-y_p## satisfies the homogeneous equation so there are values of ##C## and ##D## such that ##y_q-y_p=Cy_1+Dy_2##. This tells you that ##y_q = y_p + Cy_1+Dy_2##. Since ##y_q## could be any solution of the DE, this shows that your original ##y_p,~ y_1,~y_2## are good enough to express any solution.
     
  4. Oct 27, 2014 #3
    Thank you so much! That's exactly what I was looking for!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: How does the Wronskian determine uniqueness of solution?
  1. Unique solutions (Replies: 5)

Loading...