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Thread: cos(x)y'' + xy' - 2y = 0

  1. #1
    Junior Member
    Mar 2009

    cos(x)y'' + xy' - 2y = 0

    Find the first three non-zero terms in each of two linearly independent power series solutions centred at $\displaystyle x = 0$ of the equation

    $\displaystyle cos(x)y''+xy' - 2y = 0$

    What do you expect the radius of convergence of the solutions to be?

    I tried to right $\displaystyle cos(x)$ in its taylor series form but I got stuck somehow...$\displaystyle x=0$ is a regular singular point that I've checked...please help thanks!
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  2. #2
    Super Member
    Aug 2008
    I think zero is an ordinary point, not singular. I can derive a series solution with two arbitrary constants but can't figure out how to extract two linearly independent solutions from it. Maybe someone here can help. Writing it as:

    $\displaystyle y''+\frac{x}{\cos(x)}y'-\frac{2}{\cos(x)}y=0$


    $\displaystyle P(x)=\frac{x}{\cos(x)}=\sum_{n=0}^{\infty} b_n x^n$

    $\displaystyle Q(x)=-\frac{2}{\cos(x)}=\sum_{n=0}^{\infty} d_n x^n$

    and if we assume a solution of the form $\displaystyle y=\sum_{n=0}^{\infty} a_n x^{n+c}$

    and substitute all the sums into the DE and shift the indexes I get:

    $\displaystyle \begin{aligned}
    &\sum_{n=0}^{\infty} (n+c)(n+c-1)a_n x^{n+c-2} \\
    &\hspace{20pt}+\sum_{n=0}^{\infty}\sum_{k=0}^{n-1} (k+c)a_k b_{n-k-1}z^{n+c-2} \\
    &\hspace{20pt}+\sum_{n=0}^{\infty}\sum_{k=0}^{n-2} a_k d_{n-k-2}z^{n+c-2}=0

    collecting terms then:

    $\displaystyle c(c-1)a_0=0$

    $\displaystyle c[(1+c)a_1+a_0 b_0]=0$

    and if I choose $\displaystyle c=0$ then for $\displaystyle n\geq 2$ we have:

    $\displaystyle a_n=\frac{n}{n-1}\left(-\sum_{k=1}^{n-1} k a_k b_{n-k-1}-\sum_{k=0}^{n-2} a_k d_{n-k-2}\right)$

    and I can let $\displaystyle a_0$ and $\displaystyle a_1$ be arbitrary and it's not hard to code this in Mathematica and show at least graphically, a good fit between this series solution and a numerical solution for some IVP such as:

    $\displaystyle \cos(x)y''+x y'-2y=0,\quad y(0)=1,\quad y'(0)=0$

    The plot below superimposes a numerical solution to this IVP against the series solution using the first 20 terms. The agreement is not bad for my money.

    pseries = Series[x/Cos[x], {x, 0, 25}]; 
    Subscript[b, n_] := SeriesCoefficient[
       pseries, n]
    qseries = Series[-2/Cos[x], {x, 0, 25}]; 
    Subscript[b, n_] := SeriesCoefficient[
       pseries, n]
    Subscript[d, n_] := SeriesCoefficient[
       qseries, n]
    Subscript[a, 0] = 1; 
    Subscript[a, 1] = 0; 
    mycoef = Table[Subscript[a, n] = 
         (1/(n*(n - 1)))*
          (-Sum[k*Subscript[a, k]*Subscript[
               b, n - k - 1], {k, 1, 
              n - 1}] - Sum[Subscript[a, k]*
             Subscript[d, n - k - 2], 
            {k, 0, n - 2}]), {n, 2, 20}]; 
    myfun[x_] = Sum[Subscript[a, n]*x^n, 
       {n, 0, 20}]
    series = Plot[myfun[x], {x, 0, 2}, 
       PlotStyle -> Red]
    sol = NDSolve[
       {Cos[x]*Derivative[2][y][x] + 
          x*Derivative[1][y][x] - 2*y[x] == 
         0, y[0] == 1, Derivative[1][y][0] == 
         0}, y, {x, 0, 2}]
    numeric = Plot[Evaluate[y[x] /. sol], 
       {x, 0, 2}, PlotStyle -> Blue]
    Show[{series, numeric}]
    Attached Thumbnails Attached Thumbnails cos(x)y'' + xy' - 2y = 0-cosine-de.jpg  
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  3. #3
    Super Member
    Aug 2008
    Look,this is what I'm gonna' do with this then: I'm going to let $\displaystyle a_0=1$ and $\displaystyle a_1=0$. Then:

    $\displaystyle f_1(x)=1+x^2+\frac{x^4}{12}+\frac{x^6}{120}+\frac{ 19 x^8}{20160}+\cdots$

    Likewise, I'll let $\displaystyle a_0=0$ and $\displaystyle a_1=1$. Then:

    $\displaystyle f_2(x)=x + x^3/6 + x^5/60 + x^7/560 + x^9/4480+\cdots$

    and these are obviously linearly-independent so then I can (probably) write the solution as:

    $\displaystyle y(x)=c_1 f_1(x)+c_2 f_2(x)$

    . . . there's your five. Now, note he asked what do you "think" the radius of convergence is and not what "IS" the radius of convergence. Then that part is easy. I think the radius of convergence is $\displaystyle \pi/2$ since $\displaystyle \cos(\pi/2)=0$ and $\displaystyle P(x)$ and $\displaystyle Q(x)$ are singular there even though my plot goes past this point. And if you wanted to be more rigorous, then I'd recommend "Intermediate Differential Equations" by Rainville where I got most of this and where it is shown that this type of problem expressed as the series above has a radius of convergence at least as large as $\displaystyle \frac{r}{M}$ where this expression is derived in the text.

    If that's not it then I bet it's close . . . B is ok with me.
    Last edited by shawsend; Jan 28th 2010 at 05:47 AM.
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