Internal problem ID [1423]
Internal file name [OUTPUT/1424_Sunday_June_05_2022_02_16_29_AM_38437327/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF
FROBENIUS III. Exercises 7.7. Page 389
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
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (9-x \right ) y=0} \] 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. \[ 4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x -9\right ) y = 0 \] 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 {1}{x}\\ q(x) &= \frac {x -9}{4 x^{2}}\\ \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 \[ 4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x -9\right ) y = 0 \] 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} 4 x^{2} \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right )+4 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (x -9\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 }}4 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}4 x^{n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{1+n +r} a_{n}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-9 a_{n} x^{n +r}\right ) = 0 \end{equation} The next step is to make all powers of \(x\) be \(n +r\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{n +r}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =0}{\sum }}x^{1+n +r} a_{n} &= \moverset {\infty }{\munderset {n =1}{\sum }}a_{n -1} x^{n +r} \\ \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 \(n +r\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =0}{\sum }}4 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}4 x^{n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}a_{n -1} x^{n +r}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-9 a_{n} x^{n +r}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ 4 x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )+4 x^{n +r} a_{n} \left (n +r \right )-9 a_{n} x^{n +r} = 0 \] When \(n = 0\) the above becomes \[ 4 x^{r} a_{0} r \left (-1+r \right )+4 x^{r} a_{0} r -9 a_{0} x^{r} = 0 \] Or \[ \left (4 x^{r} r \left (-1+r \right )+4 x^{r} r -9 x^{r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (4 r^{2}-9\right ) x^{r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ 4 r^{2}-9 = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= {\frac {3}{2}}\\ r_2 &= -{\frac {3}{2}} \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \left (4 r^{2}-9\right ) x^{r} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \(\left [{\frac {3}{2}}, -{\frac {3}{2}}\right ]\).
Since \(r_1 - r_2 = 3\) 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 ) &= x^{\frac {3}{2}} \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 )+\frac {\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n}}{x^{\frac {3}{2}}} \end {align*}
Or \begin {align*} y_{1}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +\frac {3}{2}}\\ 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 -\frac {3}{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} 4 a_{n} \left (n +r \right ) \left (n +r -1\right )+4 a_{n} \left (n +r \right )+a_{n -1}-9 a_{n} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = -\frac {a_{n -1}}{4 n^{2}+8 n r +4 r^{2}-9}\tag {4} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{n} = -\frac {a_{n -1}}{4 n \left (n +3\right )}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = {\frac {3}{2}}\) 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 \[ a_{1}=-\frac {1}{4 r^{2}+8 r -5} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{1}=-{\frac {1}{16}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{4 r^{2}+8 r -5}\) | \(-{\frac {1}{16}}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {1}{16 r^{4}+96 r^{3}+136 r^{2}-24 r -35} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{2}={\frac {1}{640}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{4 r^{2}+8 r -5}\) | \(-{\frac {1}{16}}\) |
| \(a_{2}\) | \(\frac {1}{16 r^{4}+96 r^{3}+136 r^{2}-24 r -35}\) | \(\frac {1}{640}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=-\frac {1}{64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{3}=-{\frac {1}{46080}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{4 r^{2}+8 r -5}\) | \(-{\frac {1}{16}}\) |
| \(a_{2}\) | \(\frac {1}{16 r^{4}+96 r^{3}+136 r^{2}-24 r -35}\) | \(\frac {1}{640}\) |
| \(a_{3}\) | \(-\frac {1}{64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945}\) | \(-{\frac {1}{46080}}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {1}{\left (64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945\right ) \left (4 r^{2}+32 r +55\right )} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{4}={\frac {1}{5160960}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{4 r^{2}+8 r -5}\) | \(-{\frac {1}{16}}\) |
| \(a_{2}\) | \(\frac {1}{16 r^{4}+96 r^{3}+136 r^{2}-24 r -35}\) | \(\frac {1}{640}\) |
| \(a_{3}\) | \(-\frac {1}{64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945}\) | \(-{\frac {1}{46080}}\) |
| \(a_{4}\) | \(\frac {1}{\left (64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945\right ) \left (4 r^{2}+32 r +55\right )}\) | \(\frac {1}{5160960}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=-\frac {1}{\left (64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945\right ) \left (4 r^{2}+32 r +55\right ) \left (4 r^{2}+40 r +91\right )} \] Which for the root \(r = {\frac {3}{2}}\) becomes \[ a_{5}=-{\frac {1}{825753600}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(-\frac {1}{4 r^{2}+8 r -5}\) | \(-{\frac {1}{16}}\) |
| \(a_{2}\) | \(\frac {1}{16 r^{4}+96 r^{3}+136 r^{2}-24 r -35}\) | \(\frac {1}{640}\) |
| \(a_{3}\) | \(-\frac {1}{64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945}\) | \(-{\frac {1}{46080}}\) |
| \(a_{4}\) | \(\frac {1}{\left (64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945\right ) \left (4 r^{2}+32 r +55\right )}\) | \(\frac {1}{5160960}\) |
| \(a_{5}\) | \(-\frac {1}{\left (64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945\right ) \left (4 r^{2}+32 r +55\right ) \left (4 r^{2}+40 r +91\right )}\) | \(-{\frac {1}{825753600}}\) |
Using the above table, then the solution \(y_{1}\left (x \right )\) is \begin {align*} y_{1}\left (x \right )&= x^{\frac {3}{2}} \left (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}\dots \right ) \\ &= x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\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 \(N=3\). Now we need to determine if \(C\) is zero or not. This is done by finding \(\lim _{r\rightarrow r_{2}}a_{3}\left (r \right )\). 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_{3} \\ &= -\frac {1}{64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945} \end {align*}
Therefore \begin {align*} \lim _{r\rightarrow r_{2}}-\frac {1}{64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945}&= \lim _{r\rightarrow -{\frac {3}{2}}}-\frac {1}{64 r^{6}+768 r^{5}+3280 r^{4}+5760 r^{3}+2956 r^{2}-1488 r -945}\\ &= \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 \(4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x -9\right ) y = 0\) gives \[ 4 x^{2} \left (C y_{1}^{\prime \prime }\left (x \right ) \ln \left (x \right )+\frac {2 C y_{1}^{\prime }\left (x \right )}{x}-\frac {C y_{1}\left (x \right )}{x^{2}}+\moverset {\infty }{\munderset {n =0}{\sum }}\left (\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )^{2}}{x^{2}}-\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x^{2}}\right )\right )+4 x \left (C y_{1}^{\prime }\left (x \right ) \ln \left (x \right )+\frac {C y_{1}\left (x \right )}{x}+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x}\right )\right )+\left (x -9\right ) \left (C y_{1}\left (x \right ) \ln \left (x \right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +r_{2}}\right )\right ) = 0 \] Which can be written as \begin{equation} \tag{7} \left (\left (4 x^{2} y_{1}^{\prime \prime }\left (x \right )+4 y_{1}^{\prime }\left (x \right ) x +\left (x -9\right ) y_{1}\left (x \right )\right ) \ln \left (x \right )+4 x^{2} \left (\frac {2 y_{1}^{\prime }\left (x \right )}{x}-\frac {y_{1}\left (x \right )}{x^{2}}\right )+4 y_{1}\left (x \right )\right ) C +4 x^{2} \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 )+4 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x}\right )+\left (x -9\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 \[ 4 x^{2} y_{1}^{\prime \prime }\left (x \right )+4 y_{1}^{\prime }\left (x \right ) x +\left (x -9\right ) y_{1}\left (x \right ) = 0 \] Eq (7) simplifes to \begin{equation} \tag{8} \left (4 x^{2} \left (\frac {2 y_{1}^{\prime }\left (x \right )}{x}-\frac {y_{1}\left (x \right )}{x^{2}}\right )+4 y_{1}\left (x \right )\right ) C +4 x^{2} \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 )+4 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {b_{n} x^{n +r_{2}} \left (n +r_{2}\right )}{x}\right )+\left (x -9\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} 4 \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}+8 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-1+n +r_{1}} a_{n} \left (n +r_{1}\right )\right ) C +4 \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-1+n +r_{2}} b_{n} \left (n +r_{2}\right )\right ) x +\left (x -9\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +r_{2}}\right ) = 0 \end{equation} Since \(r_{1} = {\frac {3}{2}}\) and \(r_{2} = -{\frac {3}{2}}\) then the above becomes \begin{equation} \tag{10} 4 \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-\frac {7}{2}+n} b_{n} \left (n -\frac {3}{2}\right ) \left (-\frac {5}{2}+n \right )\right ) x^{2}+8 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{\frac {1}{2}+n} a_{n} \left (n +\frac {3}{2}\right )\right ) C +4 \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{-\frac {5}{2}+n} b_{n} \left (n -\frac {3}{2}\right )\right ) x +\left (x -9\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n -\frac {3}{2}}\right ) = 0 \end{equation} Expanding \(\frac {1}{\sqrt {x}}\) as Taylor series around \(x=0\) and keeping only the first \(6\) terms gives \begin {align*} \frac {1}{\sqrt {x}} &= \frac {1}{\sqrt {x}} + \dots \\ &= \frac {1}{\sqrt {x}} \end {align*}
Expanding \(-\frac {9}{x^{\frac {3}{2}}}\) as Taylor series around \(x=0\) and keeping only the first \(6\) terms gives \begin {align*} -\frac {9}{x^{\frac {3}{2}}} &= -\frac {9}{x^{\frac {3}{2}}} + \dots \\ &= -\frac {9}{x^{\frac {3}{2}}} \end {align*}
Which simplifies to \begin{equation} \tag{2A} \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n -\frac {3}{2}} b_{n} \left (4 n^{2}-16 n +15\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (8 n +12\right ) C a_{n} x^{n +\frac {3}{2}}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (4 n -6\right ) b_{n} x^{n -\frac {3}{2}}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n -\frac {1}{2}} b_{n}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-9 b_{n} x^{n -\frac {3}{2}}\right ) = 0 \end{equation} The next step is to make all powers of \(x\) be \(n -\frac {3}{2}\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{n -\frac {3}{2}}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =0}{\sum }}\left (8 n +12\right ) C a_{n} x^{n +\frac {3}{2}} &= \moverset {\infty }{\munderset {n =3}{\sum }}C a_{n -3} \left (8 n -12\right ) x^{n -\frac {3}{2}} \\ \moverset {\infty }{\munderset {n =0}{\sum }}x^{n -\frac {1}{2}} b_{n} &= \moverset {\infty }{\munderset {n =1}{\sum }}b_{n -1} x^{n -\frac {3}{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 \(n -\frac {3}{2}\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n -\frac {3}{2}} b_{n} \left (4 n^{2}-16 n +15\right )\right )+\left (\moverset {\infty }{\munderset {n =3}{\sum }}C a_{n -3} \left (8 n -12\right ) x^{n -\frac {3}{2}}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (4 n -6\right ) b_{n} x^{n -\frac {3}{2}}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}b_{n -1} x^{n -\frac {3}{2}}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-9 b_{n} x^{n -\frac {3}{2}}\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 \[ -8 b_{1}+b_{0} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ -8 b_{1}+1 = 0 \] Solving the above for \(b_{1}\) gives \[ b_{1}={\frac {1}{8}} \] For \(n=2\), Eq (2B) gives \[ -8 b_{2}+b_{1} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ -8 b_{2}+\frac {1}{8} = 0 \] Solving the above for \(b_{2}\) gives \[ b_{2}={\frac {1}{64}} \] For \(n=N\), where \(N=3\) which is the difference between the two roots, we are free to choose \(b_{3} = 0\). Hence for \(n=3\), Eq (2B) gives \[ 12 C +\frac {1}{64} = 0 \] Which is solved for \(C\). Solving for \(C\) gives \[ C=-{\frac {1}{768}} \] For \(n=4\), Eq (2B) gives \[ 20 C a_{1}+b_{3}+16 b_{4} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ 16 b_{4}+\frac {5}{3072} = 0 \] Solving the above for \(b_{4}\) gives \[ b_{4}=-{\frac {5}{49152}} \] For \(n=5\), Eq (2B) gives \[ 28 C a_{2}+b_{4}+40 b_{5} = 0 \] Which when replacing the above values found already for \(b_{n}\) and the values found earlier for \(a_{n}\) and for \(C\), gives \[ 40 b_{5}-\frac {13}{81920} = 0 \] Solving the above for \(b_{5}\) gives \[ b_{5}={\frac {13}{3276800}} \] 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 \(C=-{\frac {1}{768}}\) and all \(b_{n}\), then the second solution becomes \[ y_{2}\left (x \right )= -\frac {1}{768}\eslowast \left (x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\right )\right ) \ln \left (x \right )+\frac {1+\frac {x}{8}+\frac {x^{2}}{64}-\frac {5 x^{4}}{49152}+\frac {13 x^{5}}{3276800}+O\left (x^{6}\right )}{x^{\frac {3}{2}}} \] 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} x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\right ) + c_{2} \left (-\frac {1}{768}\eslowast \left (x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\right )\right ) \ln \left (x \right )+\frac {1+\frac {x}{8}+\frac {x^{2}}{64}-\frac {5 x^{4}}{49152}+\frac {13 x^{5}}{3276800}+O\left (x^{6}\right )}{x^{\frac {3}{2}}}\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{768}+\frac {1+\frac {x}{8}+\frac {x^{2}}{64}-\frac {5 x^{4}}{49152}+\frac {13 x^{5}}{3276800}+O\left (x^{6}\right )}{x^{\frac {3}{2}}}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{768}+\frac {1+\frac {x}{8}+\frac {x^{2}}{64}-\frac {5 x^{4}}{49152}+\frac {13 x^{5}}{3276800}+O\left (x^{6}\right )}{x^{\frac {3}{2}}}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{\frac {3}{2}} \left (1-\frac {x}{16}+\frac {x^{2}}{640}-\frac {x^{3}}{46080}+\frac {x^{4}}{5160960}-\frac {x^{5}}{825753600}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{768}+\frac {1+\frac {x}{8}+\frac {x^{2}}{64}-\frac {5 x^{4}}{49152}+\frac {13 x^{5}}{3276800}+O\left (x^{6}\right )}{x^{\frac {3}{2}}}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 x^{2} \left (\frac {d}{d x}y^{\prime }\right )+4 x y^{\prime }+\left (x -9\right ) 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 {y^{\prime }}{x}-\frac {\left (x -9\right ) y}{4 x^{2}} \\ \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 {y^{\prime }}{x}+\frac {\left (x -9\right ) y}{4 x^{2}}=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 {1}{x}, P_{3}\left (x \right )=\frac {x -9}{4 x^{2}}\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}}}=1 \\ {} & \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}}}=-\frac {9}{4} \\ {} & \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} \\ {} & {} & 4 x^{2} \left (\frac {d}{d x}y^{\prime }\right )+4 x y^{\prime }+\left (x -9\right ) 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} x^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x \cdot y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\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} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (3+2 r \right ) \left (-3+2 r \right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k} \left (2 k +2 r +3\right ) \left (2 k +2 r -3\right )+a_{k -1}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \left (3+2 r \right ) \left (-3+2 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-\frac {3}{2}, \frac {3}{2}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & a_{k} \left (4 k^{2}+8 k r +4 r^{2}-9\right )+a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & a_{k +1} \left (4 \left (k +1\right )^{2}+8 \left (k +1\right ) r +4 r^{2}-9\right )+a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {a_{k}}{4 k^{2}+8 k r +4 r^{2}+8 k +8 r -5} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {3}{2} \\ {} & {} & a_{k +1}=-\frac {a_{k}}{4 k^{2}-4 k -8} \\ \bullet & {} & \textrm {Series not valid for}\hspace {3pt} r =-\frac {3}{2}\hspace {3pt}\textrm {, division by}\hspace {3pt} 0\hspace {3pt}\textrm {in the recursion relation at}\hspace {3pt} k =2 \\ {} & {} & a_{k +1}=-\frac {a_{k}}{4 k^{2}-4 k -8} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {3}{2} \\ {} & {} & a_{k +1}=-\frac {a_{k}}{4 k^{2}+20 k +16} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {3}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {3}{2}}, a_{k +1}=-\frac {a_{k}}{4 k^{2}+20 k +16}\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: 0.046 (sec). Leaf size: 63
Order:=6; dsolve(4*x^2*diff(y(x),x$2)+4*x*diff(y(x),x)-(9-x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (1-\frac {1}{16} x +\frac {1}{640} x^{2}-\frac {1}{46080} x^{3}+\frac {1}{5160960} x^{4}-\frac {1}{825753600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {1}{64} x^{3}+\frac {1}{1024} x^{4}-\frac {1}{40960} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+\frac {3}{2} x +\frac {3}{16} x^{2}-\frac {5}{4096} x^{4}+\frac {39}{819200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 96
AsymptoticDSolveValue[4*x^2*y''[x]+4*x*y'[x]-(9-x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^{11/2}}{5160960}-\frac {x^{9/2}}{46080}+\frac {x^{7/2}}{640}-\frac {x^{5/2}}{16}+x^{3/2}\right )+c_1 \left (\frac {(x-16) x^{3/2} \log (x)}{12288}-\frac {19 x^4-64 x^3-2304 x^2-18432 x-147456}{147456 x^{3/2}}\right ) \]