Internal problem ID [6983]
Internal file name [OUTPUT/6226_Friday_August_12_2022_11_06_09_PM_76951881/index.tex]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th
edition. 1997.
Section: CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a
Positive Integer: Logarithmic Case. Exercises page 384
Problem number: 3.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Regular singular point. Difference is integer"
Maple gives the following as the ode type
[[_Emden, _Fowler]]
\[ \boxed {2 x y^{\prime \prime }+6 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ 2 x y^{\prime \prime }+6 y^{\prime }+y = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end {align*}
Where \begin {align*} p(x) &= \frac {3}{x}\\ q(x) &= \frac {1}{2 x}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([0]\)
Irregular singular points : \([\infty ]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ 2 x y^{\prime \prime }+6 y^{\prime }+y = 0 \] Let the solution be represented as Frobenius power series of the form \[ y = \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r} \] Then \begin{align*} y^{\prime } &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1} \\ y^{\prime \prime } &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2} \\ \end{align*} Substituting the above back into the ode gives \begin{equation} \tag{1} 2 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right )+6 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right ) = 0 \end{equation} Which simplifies to \begin{equation} \tag{2A} \left (\moverset {\infty }{\munderset {n =0}{\sum }}2 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}6 \left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right ) = 0 \end{equation} The next step is to make all powers of \(x\) be \(n +r -1\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{n +r -1}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r} &= \moverset {\infty }{\munderset {n =1}{\sum }}a_{n -1} x^{n +r -1} \\ \end{align*} Substituting all the above in Eq (2A) gives the following equation where now all powers of \(x\) are the same and equal to \(n +r -1\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =0}{\sum }}2 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}6 \left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}a_{n -1} x^{n +r -1}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ 2 x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )+6 \left (n +r \right ) a_{n} x^{n +r -1} = 0 \] When \(n = 0\) the above becomes \[ 2 x^{-1+r} a_{0} r \left (-1+r \right )+6 r a_{0} x^{-1+r} = 0 \] Or \[ \left (2 x^{-1+r} r \left (-1+r \right )+6 r \,x^{-1+r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ 2 r \,x^{-1+r} \left (2+r \right ) = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ 2 r \left (2+r \right ) = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= 0\\ r_2 &= -2 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ 2 r \,x^{-1+r} \left (2+r \right ) = 0 \] Solving for \(r\) gives the roots of the indicial equation as \([0, -2]\).
Since \(r_1 - r_2 = 2\) is an integer, then we can construct two linearly independent solutions \begin {align*} y_{1}\left (x \right ) &= x^{r_{1}} \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\right )\\ y_{2}\left (x \right ) &= C y_{1}\left (x \right ) \ln \left (x \right )+x^{r_{2}} \left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n}\right ) \end {align*}
Or \begin {align*} y_{1}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\\ y_{2}\left (x \right ) &= C y_{1}\left (x \right ) \ln \left (x \right )+\frac {\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n}}{x^{2}} \end {align*}
Or \begin {align*} y_{1}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\\ y_{2}\left (x \right ) &= C y_{1}\left (x \right ) \ln \left (x \right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n -2}\right ) \end {align*}
Where \(C\) above can be zero. We start by finding \(y_{1}\). Eq (2B) derived above is now used to find all \(a_{n}\) coefficients. The case \(n = 0\) is skipped since it was used to find the roots of the indicial equation. \(a_{0}\) is arbitrary and taken as \(a_{0} = 1\). For \(1\le n\) the recursive equation is \begin{equation} \tag{3} 2 a_{n} \left (n +r \right ) \left (n +r -1\right )+6 a_{n} \left (n +r \right )+a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {a_{n -1}}{2 \left (n^{2}+2 n r +r^{2}+2 n +2 r \right )}\tag {4} \] Which for the root \(r = 0\) becomes \[ a_{n} = -\frac {a_{n -1}}{2 n \left (n +2\right )}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = 0\) and after as more terms are found using the above recursive equation.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
For \(n = 1\), using the above recursive equation gives \[ a_{1}=-\frac {1}{2 r^{2}+8 r +6} \] Which for the root \(r = 0\) becomes \[ a_{1}=-{\frac {1}{6}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{2 r^{2}+8 r +6}\) | \(-{\frac {1}{6}}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {1}{4 r^{4}+40 r^{3}+140 r^{2}+200 r +96} \] Which for the root \(r = 0\) becomes \[ a_{2}={\frac {1}{96}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{2 r^{2}+8 r +6}\) | \(-{\frac {1}{6}}\) |
| \(a_{2}\) | \(\frac {1}{4 r^{4}+40 r^{3}+140 r^{2}+200 r +96}\) | \(\frac {1}{96}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=-\frac {1}{8 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right ) \left (2+r \right ) \left (r +5\right )} \] Which for the root \(r = 0\) becomes \[ a_{3}=-{\frac {1}{2880}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{2 r^{2}+8 r +6}\) | \(-{\frac {1}{6}}\) |
| \(a_{2}\) | \(\frac {1}{4 r^{4}+40 r^{3}+140 r^{2}+200 r +96}\) | \(\frac {1}{96}\) |
| \(a_{3}\) | \(-\frac {1}{8 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right ) \left (2+r \right ) \left (r +5\right )}\) | \(-{\frac {1}{2880}}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {1}{16 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right ) \left (r +6\right )} \] Which for the root \(r = 0\) becomes \[ a_{4}={\frac {1}{138240}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{2 r^{2}+8 r +6}\) | \(-{\frac {1}{6}}\) |
| \(a_{2}\) | \(\frac {1}{4 r^{4}+40 r^{3}+140 r^{2}+200 r +96}\) | \(\frac {1}{96}\) |
| \(a_{3}\) | \(-\frac {1}{8 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right ) \left (2+r \right ) \left (r +5\right )}\) | \(-{\frac {1}{2880}}\) |
| \(a_{4}\) | \(\frac {1}{16 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right ) \left (r +6\right )}\) | \(\frac {1}{138240}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=-\frac {1}{32 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right )^{2} \left (r +6\right ) \left (r +7\right )} \] Which for the root \(r = 0\) becomes \[ a_{5}=-{\frac {1}{9676800}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{2 r^{2}+8 r +6}\) | \(-{\frac {1}{6}}\) |
| \(a_{2}\) | \(\frac {1}{4 r^{4}+40 r^{3}+140 r^{2}+200 r +96}\) | \(\frac {1}{96}\) |
| \(a_{3}\) | \(-\frac {1}{8 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right ) \left (2+r \right ) \left (r +5\right )}\) | \(-{\frac {1}{2880}}\) |
| \(a_{4}\) | \(\frac {1}{16 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right ) \left (r +6\right )}\) | \(\frac {1}{138240}\) |
| \(a_{5}\) | \(-\frac {1}{32 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right )^{2} \left (r +6\right ) \left (r +7\right )}\) | \(-{\frac {1}{9676800}}\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {1}{64 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right )^{2} \left (r +6\right )^{2} \left (r +7\right ) \left (r +8\right )} \] Which for the root \(r = 0\) becomes \[ a_{6}={\frac {1}{928972800}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{2 r^{2}+8 r +6}\) | \(-{\frac {1}{6}}\) |
| \(a_{2}\) | \(\frac {1}{4 r^{4}+40 r^{3}+140 r^{2}+200 r +96}\) | \(\frac {1}{96}\) |
| \(a_{3}\) | \(-\frac {1}{8 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right ) \left (2+r \right ) \left (r +5\right )}\) | \(-{\frac {1}{2880}}\) |
| \(a_{4}\) | \(\frac {1}{16 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right ) \left (r +6\right )}\) | \(\frac {1}{138240}\) |
| \(a_{5}\) | \(-\frac {1}{32 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right )^{2} \left (r +6\right ) \left (r +7\right )}\) | \(-{\frac {1}{9676800}}\) |
| \(a_{6}\) | \(\frac {1}{64 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right )^{2} \left (r +6\right )^{2} \left (r +7\right ) \left (r +8\right )}\) | \(\frac {1}{928972800}\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=-\frac {1}{128 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right )^{2} \left (r +6\right )^{2} \left (r +7\right )^{2} \left (r +8\right ) \left (r +9\right )} \] Which for the root \(r = 0\) becomes \[ a_{7}=-{\frac {1}{117050572800}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{2 r^{2}+8 r +6}\) | \(-{\frac {1}{6}}\) |
| \(a_{2}\) | \(\frac {1}{4 r^{4}+40 r^{3}+140 r^{2}+200 r +96}\) | \(\frac {1}{96}\) |
| \(a_{3}\) | \(-\frac {1}{8 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right ) \left (2+r \right ) \left (r +5\right )}\) | \(-{\frac {1}{2880}}\) |
| \(a_{4}\) | \(\frac {1}{16 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right ) \left (r +6\right )}\) | \(\frac {1}{138240}\) |
| \(a_{5}\) | \(-\frac {1}{32 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right )^{2} \left (r +6\right ) \left (r +7\right )}\) | \(-{\frac {1}{9676800}}\) |
| \(a_{6}\) | \(\frac {1}{64 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right )^{2} \left (r +6\right )^{2} \left (r +7\right ) \left (r +8\right )}\) | \(\frac {1}{928972800}\) |
| \(a_{7}\) | \(-\frac {1}{128 \left (r +3\right )^{2} \left (r +1\right ) \left (r +4\right )^{2} \left (2+r \right ) \left (r +5\right )^{2} \left (r +6\right )^{2} \left (r +7\right )^{2} \left (r +8\right ) \left (r +9\right )}\) | \(-{\frac {1}{117050572800}}\) |
Using the above table, then the solution \(y_{1}\left (x \right )\) is \begin {align*} y_{1}\left (x \right )&= a_{0}+a_{1} x +a_{2} x^{2}+a_{3} x^{3}+a_{4} x^{4}+a_{5} x^{5}+a_{6} x^{6}+a_{7} x^{7}+a_{8} x^{8}\dots \\ &= 1-\frac {x}{6}+\frac {x^{2}}{96}-\frac {x^{3}}{2880}+\frac {x^{4}}{138240}-\frac {x^{5}}{9676800}+\frac {x^{6}}{928972800}-\frac {x^{7}}{117050572800}+O\left (x^{8}\right ) \end {align*}
Now the second solution \(y_{2}\left (x \right )\) is found. Let \[ r_{1}-r_{2} = N \] Where \(N\) is positive integer which is the difference between the two roots. \(r_{1}\) is taken as the larger root. Hence for this problem we have \(N=2\). Now we need to determine if \(C\) is zero or not. This is done by finding \(\lim _{r\rightarrow r_{2}}a_{2}\left (r \right )\). If this limit exists, then \(C = 0\), else we need to keep the log term and \(C \neq 0\). The above table shows that \begin {align*} a_N &= a_{2} \\ &= \frac {1}{4 r^{4}+40 r^{3}+140 r^{2}+200 r +96} \end {align*}
Therefore \begin {align*} \lim _{r\rightarrow r_{2}}\frac {1}{4 r^{4}+40 r^{3}+140 r^{2}+200 r +96}&= \lim _{r\rightarrow -2}\frac {1}{4 r^{4}+40 r^{3}+140 r^{2}+200 r +96}\\ &= \textit {undefined} \end {align*}
Since the limit does not exist then the log term is needed. Therefore the second solution has the form \[ y_{2}\left (x \right ) = C y_{1}\left (x \right ) \ln \left (x \right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +r_{2}}\right ) \] Therefore \begin{align*} \frac {d}{d x}y_{2}\left (x \right ) &= C y_{1}^{\prime }\left (x \right ) \ln \left (x \right )+\frac {C y_{1}\left (x \right )}{x}+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x}\right ) \\ &= C y_{1}^{\prime }\left (x \right ) \ln \left (x \right )+\frac {C y_{1}\left (x \right )}{x}+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-1+n +r_{2}} b_{n} \left (n +r_{2}\right )\right ) \\ \frac {d^{2}}{d x^{2}}y_{2}\left (x \right ) &= C y_{1}^{\prime \prime }\left (x \right ) \ln \left (x \right )+\frac {2 C y_{1}^{\prime }\left (x \right )}{x}-\frac {C y_{1}\left (x \right )}{x^{2}}+\moverset {\infty }{\munderset {n =0}{\sum }}\left (\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )^{2}}{x^{2}}-\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x^{2}}\right ) \\ &= C y_{1}^{\prime \prime }\left (x \right ) \ln \left (x \right )+\frac {2 C y_{1}^{\prime }\left (x \right )}{x}-\frac {C y_{1}\left (x \right )}{x^{2}}+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-2+n +r_{2}} b_{n} \left (n +r_{2}\right ) \left (-1+n +r_{2}\right )\right ) \\ \end{align*} Substituting these back into the given ode \(2 x y^{\prime \prime }+6 y^{\prime }+y = 0\) gives \[ 2 x \left (C y_{1}^{\prime \prime }\left (x \right ) \ln \left (x \right )+\frac {2 C y_{1}^{\prime }\left (x \right )}{x}-\frac {C y_{1}\left (x \right )}{x^{2}}+\moverset {\infty }{\munderset {n =0}{\sum }}\left (\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )^{2}}{x^{2}}-\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x^{2}}\right )\right )+6 C y_{1}^{\prime }\left (x \right ) \ln \left (x \right )+\frac {6 C y_{1}\left (x \right )}{x}+6 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x}\right )+C y_{1}\left (x \right ) \ln \left (x \right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +r_{2}}\right ) = 0 \] Which can be written as \begin{equation} \tag{7} \left (\left (2 y_{1}^{\prime \prime }\left (x \right ) x +y_{1}\left (x \right )+6 y_{1}^{\prime }\left (x \right )\right ) \ln \left (x \right )+2 x \left (\frac {2 y_{1}^{\prime }\left (x \right )}{x}-\frac {y_{1}\left (x \right )}{x^{2}}\right )+\frac {6 y_{1}\left (x \right )}{x}\right ) C +2 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )^{2}}{x^{2}}-\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x^{2}}\right )\right )+6 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +r_{2}}\right ) = 0 \end{equation} But since \(y_{1}\left (x \right )\) is a solution to the ode, then \[ 2 y_{1}^{\prime \prime }\left (x \right ) x +y_{1}\left (x \right )+6 y_{1}^{\prime }\left (x \right ) = 0 \] Eq (7) simplifes to \begin{equation} \tag{8} \left (2 x \left (\frac {2 y_{1}^{\prime }\left (x \right )}{x}-\frac {y_{1}\left (x \right )}{x^{2}}\right )+\frac {6 y_{1}\left (x \right )}{x}\right ) C +2 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )^{2}}{x^{2}}-\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x^{2}}\right )\right )+6 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +r_{2}}\right ) = 0 \end{equation} Substituting \(y_{1} = \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r_{1}}\) into the above gives \begin{equation} \tag{9} \frac {\left (4 \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-1+n +r_{1}} a_{n} \left (n +r_{1}\right )\right ) x +4 \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r_{1}}\right )\right ) C}{x}+\frac {2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-2+n +r_{2}} b_{n} \left (n +r_{2}\right ) \left (-1+n +r_{2}\right )\right ) x^{2}+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +r_{2}}\right ) x +6 \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-1+n +r_{2}} b_{n} \left (n +r_{2}\right )\right ) x}{x} = 0 \end{equation} Since \(r_{1} = 0\) and \(r_{2} = -2\) then the above becomes \begin{equation} \tag{10} \frac {\left (4 \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n -1} a_{n} n \right ) x +4 \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\right )\right ) C}{x}+\frac {2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-4+n} b_{n} \left (n -2\right ) \left (-3+n \right )\right ) x^{2}+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n -2}\right ) x +6 \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-3+n} b_{n} \left (n -2\right )\right ) x}{x} = 0 \end{equation} Which simplifies to \begin{equation} \tag{2A} \left (\moverset {\infty }{\munderset {n =0}{\sum }}4 C \,x^{n -1} a_{n} n \right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}4 C \,x^{n -1} a_{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 x^{-3+n} b_{n} \left (n -2\right ) \left (-3+n \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n -2}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}6 x^{-3+n} b_{n} \left (n -2\right )\right ) = 0 \end{equation} The next step is to make all powers of \(x\) be \(-3+n\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{-3+n}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =0}{\sum }}4 C \,x^{n -1} a_{n} n &= \moverset {\infty }{\munderset {n =2}{\sum }}4 C \left (n -2\right ) a_{n -2} x^{-3+n} \\ \moverset {\infty }{\munderset {n =0}{\sum }}4 C \,x^{n -1} a_{n} &= \moverset {\infty }{\munderset {n =2}{\sum }}4 C a_{n -2} x^{-3+n} \\ \moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n -2} &= \moverset {\infty }{\munderset {n =1}{\sum }}b_{n -1} x^{-3+n} \\ \end{align*} Substituting all the above in Eq (2A) gives the following equation where now all powers of \(x\) are the same and equal to \(-3+n\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =2}{\sum }}4 C \left (n -2\right ) a_{n -2} x^{-3+n}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}4 C a_{n -2} x^{-3+n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 x^{-3+n} b_{n} \left (n -2\right ) \left (-3+n \right )\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}b_{n -1} x^{-3+n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}6 x^{-3+n} b_{n} \left (n -2\right )\right ) = 0 \end{equation} For \(n=0\) in Eq. (2B), we choose arbitray value for \(b_{0}\) as \(b_{0} = 1\). For \(n=1\), Eq (2B) gives \[ -2 b_{1}+b_{0} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ -2 b_{1}+1 = 0 \] Solving the above for \(b_{1}\) gives \[ b_{1}={\frac {1}{2}} \] For \(n=N\), where \(N=2\) which is the difference between the two roots, we are free to choose \(b_{2} = 0\). Hence for \(n=2\), Eq (2B) gives \[ 4 C +\frac {1}{2} = 0 \] Which is solved for \(C\). Solving for \(C\) gives \[ C=-{\frac {1}{8}} \] For \(n=3\), Eq (2B) gives \[ 8 C a_{1}+b_{2}+6 b_{3} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ 6 b_{3}+\frac {1}{6} = 0 \] Solving the above for \(b_{3}\) gives \[ b_{3}=-{\frac {1}{36}} \] For \(n=4\), Eq (2B) gives \[ 12 C a_{2}+b_{3}+16 b_{4} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ 16 b_{4}-\frac {25}{576} = 0 \] Solving the above for \(b_{4}\) gives \[ b_{4}={\frac {25}{9216}} \] For \(n=5\), Eq (2B) gives \[ 16 C a_{3}+b_{4}+30 b_{5} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ 30 b_{5}+\frac {157}{46080} = 0 \] Solving the above for \(b_{5}\) gives \[ b_{5}=-{\frac {157}{1382400}} \] For \(n=6\), Eq (2B) gives \[ 20 C a_{4}+b_{5}+48 b_{6} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ 48 b_{6}-\frac {91}{691200} = 0 \] Solving the above for \(b_{6}\) gives \[ b_{6}={\frac {91}{33177600}} \] For \(n=7\), Eq (2B) gives \[ 24 C a_{5}+b_{6}+70 b_{7} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ 70 b_{7}+\frac {709}{232243200} = 0 \] Solving the above for \(b_{7}\) gives \[ b_{7}=-{\frac {709}{16257024000}} \] Now that we found all \(b_{n}\) and \(C\), we can calculate the second solution from \[ y_{2}\left (x \right ) = C y_{1}\left (x \right ) \ln \left (x \right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +r_{2}}\right ) \] Using the above value found for \(C=-{\frac {1}{8}}\) and all \(b_{n}\), then the second solution becomes \[ y_{2}\left (x \right )= \left (-\frac {1}{8}\eslowast \left (1-\frac {x}{6}+\frac {x^{2}}{96}-\frac {x^{3}}{2880}+\frac {x^{4}}{138240}-\frac {x^{5}}{9676800}+\frac {x^{6}}{928972800}-\frac {x^{7}}{117050572800}+O\left (x^{8}\right )\right )\right ) \ln \left (x \right )+\frac {1+\frac {x}{2}-\frac {x^{3}}{36}+\frac {25 x^{4}}{9216}-\frac {157 x^{5}}{1382400}+\frac {91 x^{6}}{33177600}-\frac {709 x^{7}}{16257024000}+O\left (x^{8}\right )}{x^{2}} \] Therefore the homogeneous solution is \begin{align*} y_h(x) &= c_{1} y_{1}\left (x \right )+c_{2} y_{2}\left (x \right ) \\ &= c_{1} \left (1-\frac {x}{6}+\frac {x^{2}}{96}-\frac {x^{3}}{2880}+\frac {x^{4}}{138240}-\frac {x^{5}}{9676800}+\frac {x^{6}}{928972800}-\frac {x^{7}}{117050572800}+O\left (x^{8}\right )\right ) + c_{2} \left (\left (-\frac {1}{8}\eslowast \left (1-\frac {x}{6}+\frac {x^{2}}{96}-\frac {x^{3}}{2880}+\frac {x^{4}}{138240}-\frac {x^{5}}{9676800}+\frac {x^{6}}{928972800}-\frac {x^{7}}{117050572800}+O\left (x^{8}\right )\right )\right ) \ln \left (x \right )+\frac {1+\frac {x}{2}-\frac {x^{3}}{36}+\frac {25 x^{4}}{9216}-\frac {157 x^{5}}{1382400}+\frac {91 x^{6}}{33177600}-\frac {709 x^{7}}{16257024000}+O\left (x^{8}\right )}{x^{2}}\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} \left (1-\frac {x}{6}+\frac {x^{2}}{96}-\frac {x^{3}}{2880}+\frac {x^{4}}{138240}-\frac {x^{5}}{9676800}+\frac {x^{6}}{928972800}-\frac {x^{7}}{117050572800}+O\left (x^{8}\right )\right )+c_{2} \left (\left (-\frac {1}{8}+\frac {x}{48}-\frac {x^{2}}{768}+\frac {x^{3}}{23040}-\frac {x^{4}}{1105920}+\frac {x^{5}}{77414400}-\frac {x^{6}}{7431782400}+\frac {x^{7}}{936404582400}-\frac {O\left (x^{8}\right )}{8}\right ) \ln \left (x \right )+\frac {1+\frac {x}{2}-\frac {x^{3}}{36}+\frac {25 x^{4}}{9216}-\frac {157 x^{5}}{1382400}+\frac {91 x^{6}}{33177600}-\frac {709 x^{7}}{16257024000}+O\left (x^{8}\right )}{x^{2}}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \left (1-\frac {x}{6}+\frac {x^{2}}{96}-\frac {x^{3}}{2880}+\frac {x^{4}}{138240}-\frac {x^{5}}{9676800}+\frac {x^{6}}{928972800}-\frac {x^{7}}{117050572800}+O\left (x^{8}\right )\right )+c_{2} \left (\left (-\frac {1}{8}+\frac {x}{48}-\frac {x^{2}}{768}+\frac {x^{3}}{23040}-\frac {x^{4}}{1105920}+\frac {x^{5}}{77414400}-\frac {x^{6}}{7431782400}+\frac {x^{7}}{936404582400}-\frac {O\left (x^{8}\right )}{8}\right ) \ln \left (x \right )+\frac {1+\frac {x}{2}-\frac {x^{3}}{36}+\frac {25 x^{4}}{9216}-\frac {157 x^{5}}{1382400}+\frac {91 x^{6}}{33177600}-\frac {709 x^{7}}{16257024000}+O\left (x^{8}\right )}{x^{2}}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \left (1-\frac {x}{6}+\frac {x^{2}}{96}-\frac {x^{3}}{2880}+\frac {x^{4}}{138240}-\frac {x^{5}}{9676800}+\frac {x^{6}}{928972800}-\frac {x^{7}}{117050572800}+O\left (x^{8}\right )\right )+c_{2} \left (\left (-\frac {1}{8}+\frac {x}{48}-\frac {x^{2}}{768}+\frac {x^{3}}{23040}-\frac {x^{4}}{1105920}+\frac {x^{5}}{77414400}-\frac {x^{6}}{7431782400}+\frac {x^{7}}{936404582400}-\frac {O\left (x^{8}\right )}{8}\right ) \ln \left (x \right )+\frac {1+\frac {x}{2}-\frac {x^{3}}{36}+\frac {25 x^{4}}{9216}-\frac {157 x^{5}}{1382400}+\frac {91 x^{6}}{33177600}-\frac {709 x^{7}}{16257024000}+O\left (x^{8}\right )}{x^{2}}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x \left (\frac {d}{d x}y^{\prime }\right )+6 y^{\prime }+y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=-\frac {3 y^{\prime }}{x}-\frac {y}{2 x} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }+\frac {3 y^{\prime }}{x}+\frac {y}{2 x}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {3}{x}, P_{3}\left (x \right )=\frac {1}{2 x}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=3 \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=0 \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & 2 x \left (\frac {d}{d x}y^{\prime }\right )+6 y^{\prime }+y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & y^{\prime }=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x \cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & x \cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) \left (k +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & 2 a_{0} r \left (2+r \right ) x^{-1+r}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (2 a_{k +1} \left (k +1+r \right ) \left (k +3+r \right )+a_{k}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & 2 r \left (2+r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-2, 0\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & 2 a_{k +1} \left (k +1+r \right ) \left (k +3+r \right )+a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {a_{k}}{2 \left (k +1+r \right ) \left (k +3+r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-2 \\ {} & {} & a_{k +1}=-\frac {a_{k}}{2 \left (k -1\right ) \left (k +1\right )} \\ \bullet & {} & \textrm {Series not valid for}\hspace {3pt} r =-2\hspace {3pt}\textrm {, division by}\hspace {3pt} 0\hspace {3pt}\textrm {in the recursion relation at}\hspace {3pt} k =1 \\ {} & {} & a_{k +1}=-\frac {a_{k}}{2 \left (k -1\right ) \left (k +1\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +1}=-\frac {a_{k}}{2 \left (k +1\right ) \left (k +3\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +1}=-\frac {a_{k}}{2 \left (k +1\right ) \left (k +3\right )}\right ] \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 74
Order:=8; dsolve(2*x*diff(y(x),x$2)+6*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{6} x +\frac {1}{96} x^{2}-\frac {1}{2880} x^{3}+\frac {1}{138240} x^{4}-\frac {1}{9676800} x^{5}+\frac {1}{928972800} x^{6}-\frac {1}{117050572800} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{384} x^{4}-\frac {1}{11520} x^{5}+\frac {1}{552960} x^{6}-\frac {1}{38707200} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-x +\frac {1}{18} x^{3}-\frac {25}{4608} x^{4}+\frac {157}{691200} x^{5}-\frac {91}{16588800} x^{6}+\frac {709}{8128512000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.046 (sec). Leaf size: 114
AsymptoticDSolveValue[2*x*y''[x]+6*y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {x^6}{928972800}-\frac {x^5}{9676800}+\frac {x^4}{138240}-\frac {x^3}{2880}+\frac {x^2}{96}-\frac {x}{6}+1\right )+c_1 \left (\frac {53 x^6-2244 x^5+55800 x^4-633600 x^3+1036800 x^2+8294400 x+16588800}{16588800 x^2}-\frac {\left (x^4-48 x^3+1440 x^2-23040 x+138240\right ) \log (x)}{1105920}\right ) \]