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

Is this behaviour generalizable?

  1. Mar 6, 2014 #1
    Notice that, when solving second-order linear homogeneous constant-coefficient ODEs, the classic linearly independent solutions are ##y_1=e^{r_1\cdot t}## and ##y_2=e^{r_2\cdot t}## for roots ##r_1,r_2## of our auxiliary equation. (Of course, we can pick any other basis for this two-dimensional vector space.)

    If those are linearly dependent, the independent solutions chosen are ##y_1=e^{r\cdot t}## (no surprise) and ##y_2=t\cdot y_1=\frac{\partial y_1}{\partial r}##.

    Similarly, for homogeneous second-order (insert long list of adjectives here) Cauchy-Euler equations, we set up a quadratic equation and find ##y_1=t^{r_1}## and ##y_2=t^{r_2}## as our linearly independent solutions, or, if those are linearly dependent, ##y_1=t^r## and ##y_2=t^r\cdot\log\left(t\right)=\frac{\partial y_1}{\partial r}##.

    This is suggesting that, for nth-order linear homogeneous ODEs, if we can guess solutions of the form ##f\left(r,t\right)##,

    If n solutions ##r_1,\ldots,r_n## show up for ##r##, we can use ##\left\{f\left(r_1,t\right),\ldots,f\left(r_n,t\right)\right\}## as a basis for the set of solutions.

    If any repeated roots show up (slightly sketchy since it might not be a polynomial in terms of ##r##,) we use ##f\left(r_1,t\right),\ldots,f\left(r_n,t\right)## along with the partial derivatives with respect to ##r_k##, up to the ##\mathrm{mult}\left(r_k\right)##th derivative, for all ##k##, as our basis.

    Now, for obvious reasons, this probably only holds in specific situations. Have I run into the specific couple for which our linearly independent solutions are generated by taking the partial derivative if a repeated root shows up, or is this generalizable in any way such as the near-unreadable example I suggested?
     
  2. jcsd
  3. Mar 6, 2014 #2

    lurflurf

    User Avatar
    Homework Helper

    How generalizable?
    Cauchy-Euler equations are related to constant-coefficient by the chain rule.
    $$\left( \dfrac{dx}{du} \dfrac{d}{dx} \right)^n=\left( \dfrac{d}{du} \right)^n$$
    so if u=g(x)
    $$\mathrm{y}_2(u)=u \, \mathrm{y}_1(u)$$
    then
    $$\mathrm{y}_2(\mathrm{g}(x))=\mathrm{g}(x) \, \mathrm{y}_1(\mathrm{g}(x))$$
    where in your examples
    u=log(x) for Cauchy-Euler equations
    u=x for constant-coefficient
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Is this behaviour generalizable?
  1. Asymptotic behaviour (Replies: 2)

Loading...