Internal problem ID [6987]
Internal file name [OUTPUT/6230_Friday_August_12_2022_11_06_19_PM_66811311/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: 7.
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
[_Jacobi]
\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+2 \left (1-x \right ) y^{\prime }+2 y=0} \] 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. \[ \left (-x^{2}+x \right ) y^{\prime \prime }+\left (-2 x +2\right ) y^{\prime }+2 y = 0 \] 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 {2}{x}\\ q(x) &= -\frac {2}{x \left (-1+x \right )}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([0, 1, \infty ]\)
Irregular singular points : \([]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ -y^{\prime \prime } x \left (-1+x \right )+\left (-2 x +2\right ) y^{\prime }+2 y = 0 \] 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} -\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right ) x \left (-1+x \right )+\left (-2 x +2\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right ) = 0 \end{equation} Which simplifies to \begin{equation} \tag{2A} \moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 x^{n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 \left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 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 }}\left (-x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \left (n +r -1\right ) \left (n +r -2\right ) x^{n +r -1}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 x^{n +r} a_{n} \left (n +r \right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-2 a_{n -1} \left (n +r -1\right ) x^{n +r -1}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}2 a_{n} x^{n +r} &= \moverset {\infty }{\munderset {n =1}{\sum }}2 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} \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \left (n +r -1\right ) \left (n +r -2\right ) x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-2 a_{n -1} \left (n +r -1\right ) x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 \left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}2 a_{n -1} x^{n +r -1}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ x^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )+2 \left (n +r \right ) a_{n} x^{n +r -1} = 0 \] When \(n = 0\) the above becomes \[ x^{-1+r} a_{0} r \left (-1+r \right )+2 r a_{0} x^{-1+r} = 0 \] Or \[ \left (x^{-1+r} r \left (-1+r \right )+2 r \,x^{-1+r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ r \,x^{-1+r} \left (1+r \right ) = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ r \left (1+r \right ) = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= 0\\ r_2 &= -1 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ r \,x^{-1+r} \left (1+r \right ) = 0 \] Solving for \(r\) gives the roots of the indicial equation as \([0, -1]\).
Since \(r_1 - r_2 = 1\) 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} \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 -1}\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} -a_{n -1} \left (n +r -1\right ) \left (n +r -2\right )+a_{n} \left (n +r \right ) \left (n +r -1\right )-2 a_{n -1} \left (n +r -1\right )+2 a_{n} \left (n +r \right )+2 a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = \frac {\left (n +r -2\right ) a_{n -1}}{n +r}\tag {4} \] Which for the root \(r = 0\) becomes \[ a_{n} = \frac {\left (n -2\right ) a_{n -1}}{n}\tag {5} \] 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 \[ a_{1}=\frac {-1+r}{1+r} \] Which for the root \(r = 0\) becomes \[ a_{1}=-1 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-1+r}{1+r}\) | \(-1\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {\left (-1+r \right ) r}{\left (1+r \right ) \left (2+r \right )} \] Which for the root \(r = 0\) becomes \[ a_{2}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-1+r}{1+r}\) | \(-1\) |
| \(a_{2}\) | \(\frac {\left (-1+r \right ) r}{\left (1+r \right ) \left (2+r \right )}\) | \(0\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {r \left (-1+r \right )}{\left (2+r \right ) \left (3+r \right )} \] Which for the root \(r = 0\) becomes \[ a_{3}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-1+r}{1+r}\) | \(-1\) |
| \(a_{2}\) | \(\frac {\left (-1+r \right ) r}{\left (1+r \right ) \left (2+r \right )}\) | \(0\) |
| \(a_{3}\) | \(\frac {r \left (-1+r \right )}{\left (2+r \right ) \left (3+r \right )}\) | \(0\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {\left (-1+r \right ) r}{\left (3+r \right ) \left (4+r \right )} \] Which for the root \(r = 0\) becomes \[ a_{4}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-1+r}{1+r}\) | \(-1\) |
| \(a_{2}\) | \(\frac {\left (-1+r \right ) r}{\left (1+r \right ) \left (2+r \right )}\) | \(0\) |
| \(a_{3}\) | \(\frac {r \left (-1+r \right )}{\left (2+r \right ) \left (3+r \right )}\) | \(0\) |
| \(a_{4}\) | \(\frac {\left (-1+r \right ) r}{\left (3+r \right ) \left (4+r \right )}\) | \(0\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {r \left (-1+r \right )}{\left (4+r \right ) \left (5+r \right )} \] Which for the root \(r = 0\) becomes \[ a_{5}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-1+r}{1+r}\) | \(-1\) |
| \(a_{2}\) | \(\frac {\left (-1+r \right ) r}{\left (1+r \right ) \left (2+r \right )}\) | \(0\) |
| \(a_{3}\) | \(\frac {r \left (-1+r \right )}{\left (2+r \right ) \left (3+r \right )}\) | \(0\) |
| \(a_{4}\) | \(\frac {\left (-1+r \right ) r}{\left (3+r \right ) \left (4+r \right )}\) | \(0\) |
| \(a_{5}\) | \(\frac {r \left (-1+r \right )}{\left (4+r \right ) \left (5+r \right )}\) | \(0\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {r \left (-1+r \right )}{\left (5+r \right ) \left (6+r \right )} \] Which for the root \(r = 0\) becomes \[ a_{6}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-1+r}{1+r}\) | \(-1\) |
| \(a_{2}\) | \(\frac {\left (-1+r \right ) r}{\left (1+r \right ) \left (2+r \right )}\) | \(0\) |
| \(a_{3}\) | \(\frac {r \left (-1+r \right )}{\left (2+r \right ) \left (3+r \right )}\) | \(0\) |
| \(a_{4}\) | \(\frac {\left (-1+r \right ) r}{\left (3+r \right ) \left (4+r \right )}\) | \(0\) |
| \(a_{5}\) | \(\frac {r \left (-1+r \right )}{\left (4+r \right ) \left (5+r \right )}\) | \(0\) |
| \(a_{6}\) | \(\frac {r \left (-1+r \right )}{\left (5+r \right ) \left (6+r \right )}\) | \(0\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=\frac {r \left (-1+r \right )}{\left (6+r \right ) \left (7+r \right )} \] Which for the root \(r = 0\) becomes \[ a_{7}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-1+r}{1+r}\) | \(-1\) |
| \(a_{2}\) | \(\frac {\left (-1+r \right ) r}{\left (1+r \right ) \left (2+r \right )}\) | \(0\) |
| \(a_{3}\) | \(\frac {r \left (-1+r \right )}{\left (2+r \right ) \left (3+r \right )}\) | \(0\) |
| \(a_{4}\) | \(\frac {\left (-1+r \right ) r}{\left (3+r \right ) \left (4+r \right )}\) | \(0\) |
| \(a_{5}\) | \(\frac {r \left (-1+r \right )}{\left (4+r \right ) \left (5+r \right )}\) | \(0\) |
| \(a_{6}\) | \(\frac {r \left (-1+r \right )}{\left (5+r \right ) \left (6+r \right )}\) | \(0\) |
| \(a_{7}\) | \(\frac {r \left (-1+r \right )}{\left (6+r \right ) \left (7+r \right )}\) | \(0\) |
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-x +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 \(N=1\). Now we need to determine if \(C\) is zero or not. This is done by finding \(\lim _{r\rightarrow r_{2}}a_{1}\left (r \right )\). 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_{1} \\ &= \frac {-1+r}{1+r} \end {align*}
Therefore \begin {align*} \lim _{r\rightarrow r_{2}}\frac {-1+r}{1+r}&= \lim _{r\rightarrow -1}\frac {-1+r}{1+r}\\ &= \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 \(-y^{\prime \prime } x \left (-1+x \right )+\left (-2 x +2\right ) y^{\prime }+2 y = 0\) gives \[ -\left (C y_{1}^{\prime \prime }\left (x \right ) \ln \left (x \right )+\frac {2 C y_{1}^{\prime }\left (x \right )}{x}-\frac {C y_{1}\left (x \right )}{x^{2}}+\moverset {\infty }{\munderset {n =0}{\sum }}\left (\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )^{2}}{x^{2}}-\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x^{2}}\right )\right ) x \left (-1+x \right )+\left (-2 x +2\right ) \left (C y_{1}^{\prime }\left (x \right ) \ln \left (x \right )+\frac {C y_{1}\left (x \right )}{x}+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x}\right )\right )+2 C y_{1}\left (x \right ) \ln \left (x \right )+2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +r_{2}}\right ) = 0 \] Which can be written as \begin{equation} \tag{7} \left (\left (-y_{1}^{\prime \prime }\left (x \right ) x \left (-1+x \right )+\left (-2 x +2\right ) y_{1}^{\prime }\left (x \right )+2 y_{1}\left (x \right )\right ) \ln \left (x \right )-\left (\frac {2 y_{1}^{\prime }\left (x \right )}{x}-\frac {y_{1}\left (x \right )}{x^{2}}\right ) x \left (-1+x \right )+\frac {\left (-2 x +2\right ) y_{1}\left (x \right )}{x}\right ) C -\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 ) x \left (-1+x \right )+\left (-2 x +2\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x}\right )+2 \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 \[ -y_{1}^{\prime \prime }\left (x \right ) x \left (-1+x \right )+\left (-2 x +2\right ) y_{1}^{\prime }\left (x \right )+2 y_{1}\left (x \right ) = 0 \] Eq (7) simplifes to \begin{equation} \tag{8} \left (-\left (\frac {2 y_{1}^{\prime }\left (x \right )}{x}-\frac {y_{1}\left (x \right )}{x^{2}}\right ) x \left (-1+x \right )+\frac {\left (-2 x +2\right ) y_{1}\left (x \right )}{x}\right ) C -\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 ) x \left (-1+x \right )+\left (-2 x +2\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x}\right )+2 \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 (-2 x \left (-1+x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-1+n +r_{1}} a_{n} \left (n +r_{1}\right )\right )-\left (-1+x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r_{1}}\right )\right ) C}{x}+\frac {\left (-x^{3}+x^{2}\right ) \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 )+\left (-2 x^{2}+2 x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-1+n +r_{2}} b_{n} \left (n +r_{2}\right )\right )+2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +r_{2}}\right ) x}{x} = 0 \end{equation} Since \(r_{1} = 0\) and \(r_{2} = -1\) then the above becomes \begin{equation} \tag{10} \frac {\left (-2 x \left (-1+x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n -1} a_{n} n \right )-\left (-1+x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\right )\right ) C}{x}+\frac {\left (-x^{3}+x^{2}\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-3+n} b_{n} \left (n -1\right ) \left (n -2\right )\right )+\left (-2 x^{2}+2 x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n -2} b_{n} \left (n -1\right )\right )+2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n -1}\right ) x}{x} = 0 \end{equation} Which simplifies to \begin{equation} \tag{2A} \left (\moverset {\infty }{\munderset {n =0}{\sum }}2 C n \,x^{n -1} a_{n}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 C \,x^{n} a_{n} n \right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}C \,x^{n -1} a_{n}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-C a_{n} x^{n}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-b_{n} x^{n -1} \left (n -1\right ) \left (n -2\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n -2} b_{n} \left (n^{2}-3 n +2\right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 x^{n -1} b_{n} \left (n -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 x^{n -2} b_{n} \left (n -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 b_{n} x^{n -1}\right ) = 0 \end{equation} The next step is to make all powers of \(x\) be \(n -2\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{n -2}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =0}{\sum }}2 C n \,x^{n -1} a_{n} &= \moverset {\infty }{\munderset {n =1}{\sum }}2 C \left (n -1\right ) a_{n -1} x^{n -2} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 C \,x^{n} a_{n} n \right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}\left (-2 C \left (n -2\right ) a_{n -2} x^{n -2}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}C \,x^{n -1} a_{n} &= \moverset {\infty }{\munderset {n =1}{\sum }}C a_{n -1} x^{n -2} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-C a_{n} x^{n}\right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}\left (-C a_{n -2} x^{n -2}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-b_{n} x^{n -1} \left (n -1\right ) \left (n -2\right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-b_{n -1} \left (-3+n \right ) \left (n -2\right ) x^{n -2}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 x^{n -1} b_{n} \left (n -1\right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-2 b_{n -1} \left (n -2\right ) x^{n -2}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}2 b_{n} x^{n -1} &= \moverset {\infty }{\munderset {n =1}{\sum }}2 b_{n -1} x^{n -2} \\ \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 \(n -2\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =1}{\sum }}2 C \left (n -1\right ) a_{n -1} x^{n -2}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-2 C \left (n -2\right ) a_{n -2} x^{n -2}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}C a_{n -1} x^{n -2}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-C a_{n -2} x^{n -2}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-b_{n -1} \left (-3+n \right ) \left (n -2\right ) x^{n -2}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n -2} b_{n} \left (n^{2}-3 n +2\right )\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-2 b_{n -1} \left (n -2\right ) x^{n -2}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 x^{n -2} b_{n} \left (n -1\right )\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}2 b_{n -1} x^{n -2}\right ) = 0 \end{equation} For \(n=0\) in Eq. (2B), we choose arbitray value for \(b_{0}\) as \(b_{0} = 1\). For \(n=N\), where \(N=1\) which is the difference between the two roots, we are free to choose \(b_{1} = 0\). Hence for \(n=1\), Eq (2B) gives \[ C +2 = 0 \] Which is solved for \(C\). Solving for \(C\) gives \[ C=-2 \] For \(n=2\), Eq (2B) gives \[ \left (-a_{0}+3 a_{1}\right ) C +2 b_{1}+2 b_{2} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ 8+2 b_{2} = 0 \] Solving the above for \(b_{2}\) gives \[ b_{2}=-4 \] For \(n=3\), Eq (2B) gives \[ \left (-3 a_{1}+5 a_{2}\right ) C +6 b_{3} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ -6+6 b_{3} = 0 \] Solving the above for \(b_{3}\) gives \[ b_{3}=1 \] For \(n=4\), Eq (2B) gives \[ \left (-5 a_{2}+7 a_{3}\right ) C -4 b_{3}+12 b_{4} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ -4+12 b_{4} = 0 \] Solving the above for \(b_{4}\) gives \[ b_{4}={\frac {1}{3}} \] For \(n=5\), Eq (2B) gives \[ \left (-7 a_{3}+9 a_{4}\right ) C -10 b_{4}+20 b_{5} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ -\frac {10}{3}+20 b_{5} = 0 \] Solving the above for \(b_{5}\) gives \[ b_{5}={\frac {1}{6}} \] For \(n=6\), Eq (2B) gives \[ \left (-9 a_{4}+11 a_{5}\right ) C -18 b_{5}+30 b_{6} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ -3+30 b_{6} = 0 \] Solving the above for \(b_{6}\) gives \[ b_{6}={\frac {1}{10}} \] For \(n=7\), Eq (2B) gives \[ \left (-11 a_{5}+13 a_{6}\right ) C -28 b_{6}+42 b_{7} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ -\frac {14}{5}+42 b_{7} = 0 \] Solving the above for \(b_{7}\) gives \[ b_{7}={\frac {1}{15}} \] 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 \(C=-2\) and all \(b_{n}\), then the second solution becomes \[ y_{2}\left (x \right )= \left (\left (-2\right )\eslowast \left (1-x +O\left (x^{8}\right )\right )\right ) \ln \left (x \right )+\frac {1-4 x^{2}+x^{3}+\frac {x^{4}}{3}+\frac {x^{5}}{6}+\frac {x^{6}}{10}+\frac {x^{7}}{15}+O\left (x^{8}\right )}{x} \] 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-x +O\left (x^{8}\right )\right ) + c_{2} \left (\left (\left (-2\right )\eslowast \left (1-x +O\left (x^{8}\right )\right )\right ) \ln \left (x \right )+\frac {1-4 x^{2}+x^{3}+\frac {x^{4}}{3}+\frac {x^{5}}{6}+\frac {x^{6}}{10}+\frac {x^{7}}{15}+O\left (x^{8}\right )}{x}\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} \left (1-x +O\left (x^{8}\right )\right )+c_{2} \left (\left (-2+2 x -2 O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {1-4 x^{2}+x^{3}+\frac {x^{4}}{3}+\frac {x^{5}}{6}+\frac {x^{6}}{10}+\frac {x^{7}}{15}+O\left (x^{8}\right )}{x}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \left (1-x +O\left (x^{8}\right )\right )+c_{2} \left (\left (-2+2 x -2 O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {1-4 x^{2}+x^{3}+\frac {x^{4}}{3}+\frac {x^{5}}{6}+\frac {x^{6}}{10}+\frac {x^{7}}{15}+O\left (x^{8}\right )}{x}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \left (1-x +O\left (x^{8}\right )\right )+c_{2} \left (\left (-2+2 x -2 O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {1-4 x^{2}+x^{3}+\frac {x^{4}}{3}+\frac {x^{5}}{6}+\frac {x^{6}}{10}+\frac {x^{7}}{15}+O\left (x^{8}\right )}{x}\right ) \] Verified OK.
Maple trace Kovacic algorithm successful
`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 A Liouvillian solution exists Reducible group (found an exponential solution) Group is reducible, not completely reducible <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 54
Order:=8; dsolve(x*(1-x)*diff(y(x),x$2)+2*(1-x)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {\ln \left (x \right ) \left (\left (-2\right ) x +2 x^{2}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} +c_{1} \left (1-x +\operatorname {O}\left (x^{8}\right )\right ) x +\left (1-4 x^{2}+x^{3}+\frac {1}{3} x^{4}+\frac {1}{6} x^{5}+\frac {1}{10} x^{6}+\frac {1}{15} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]
✓ Solution by Mathematica
Time used: 0.408 (sec). Leaf size: 60
AsymptoticDSolveValue[x*(1-x)*y''[x]+2*(1-x)*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {3 x^6+5 x^5+10 x^4+30 x^3-150 x^2+30 x+30}{30 x}+2 (x-1) \log (x)\right )+c_2 (1-x) \]