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Section 5.2 – Series Solutions Near an Ordinary Point I

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 
  1. For the equation \((x^2+1)y''+xy'-y=0\)
    1. Determine a lower bound for the radius of convergence of the series solutions of the differential equation about \(x_0=0\).
    2. Seek its power series solution about \(x_0=0\); find the recurrence relation.
    3. Find the general term of each solution \(y_1(x)\) and \(y_2(x)\).
    4. Find the first four terms in each of two solutions \(y_1\) and \(y_2\). Show that \(W[y_1, y_2](0)\ne 0\).​

      1. The lower bound is 1.
      2. \(\displaystyle \sum_{n=0}^{\infty}n(n-1)a_nx^n+\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n+\sum_{n=0}^{\infty}na_nx^n-\sum_{n=0}^{\infty}a_nx^n=0\) . The recurrence relation is \((n+2)a_{n+2}+(n-1)a_n=0\) for all \(n=0,1,2,..\)
      3. \(y_1=1+\displaystyle\sum_{k=1}^{\infty}(-1)^{k-1}\dfrac{(2k-3)!!}{(2k)!!}x^{2k}\) where \((-1)!!=1, \quad\) \(y_2=x\)
      4. \(y_1=1+\dfrac{1}{2}x^2-\dfrac{1}{(2)(4)}x^4+\dfrac{3(1)}{(2)(4)(6)}x^6-...\quad\) and \(\quad\quad y_2=x\)
        The Wronskian is \[W[y_1,y_2](0)= \begin{vmatrix}
            1 & 0 \\ 
            0 & 1  \\ 
          \end{vmatrix}
          =1
        \]

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V71


  2. For the equation \((x^2+1)y''-6y=0\)
    1. Determine a lower bound for the radius of convergence of the series solutions of the differential equation about \(x_0=0\).
    2. Seek its power series solution about \(x_0=0\); find the recurrence relation.
    3. Find the general term of each solution \(y_1(x)\) and \(y_2(x)\).
    4. Find the first four terms in each of two solutions \(y_1\) and \(y_2\). Show that \(W[y_1, y_2](0)\ne 0\).​

      1. The lower bound is 1.
      2. \(\displaystyle \sum_{n=0}^{\infty}n(n-1)a_nx^n+\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n-\sum_{n=0}^{\infty}6a_nx^n=0\) . The recurrence relation is \(a_{n+2}=-\dfrac{n-3}{n+1}a_n\) for all \(n=0,1,2,...\)
      3. \(y_1=1+3x^2+x^4+\displaystyle\sum_{k=3}^{\infty}\dfrac{3(-1)^k}{(2k-1)(2k-3)}x^{2k}, \quad \quad\) \(y_2=x+x^3\)
      4. \(y_1=1+3x^2+x^4-\dfrac{1}{5}x^6+... \quad \quad\) \(y_2=x+x^3\)
        The Wronskian is \[
          W[y_1,y_2](0)=
          \begin{vmatrix}
            1 & 0 \\ 
            0 & 1  \\ 
          \end{vmatrix}
          =1
        \]

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V72


  3. Seek a power series solution for the initial value problem \[y''-(1+x)y=0,\quad y(0)=1,y'(0)=-2\] up to the terms of degree 5. Then do the same for finding the general solution of the equation.​

    \[y=1-2x+\dfrac{1}{2}x^2-\dfrac{1}{6}x^3-\dfrac{1}{8}x^4+\dfrac{1}{60}x^5+...\quad \]
    For the general solution of the equation, we have that \(y=a_0y_1+a_1y_2\), where \[y_1=1+\dfrac{1}{2}x^2+\dfrac{1}{6}x^3+\dfrac{1}{24}x^4+\dfrac{1}{30}x^5+... \quad\] and \(\quad\) \[y_2=x+\dfrac{1}{6}x^3+\dfrac{1}{12}x^4+\dfrac{1}{120}x^5+...\]


    To see the full video page and find related videos, click the following link.
    MATH 308 WIR22A V73