Internal problem ID [6464]
Internal file name [OUTPUT/5712_Sunday_June_05_2022_03_48_19_PM_62898894/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Section 4.5. More on Regular
Singular Points. Page 183
Problem number: 3(b).
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]]
\[ \boxed {x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x^{2}-2\right ) y=0} \] 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. \[ x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x^{2}-2\right ) y = 0 \] 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) &= -1\\ q(x) &= \frac {x^{2}-2}{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 \[ x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x^{2}-2\right ) y = 0 \] 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} 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 )-x^{2} \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (x^{2}-2\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 }}x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{1+n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r +2} a_{n}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 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 }}\left (-x^{1+n +r} a_{n} \left (n +r \right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \left (n +r -1\right ) x^{n +r}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r +2} a_{n} &= \moverset {\infty }{\munderset {n =2}{\sum }}a_{n -2} 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 }}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 ) x^{n +r}\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}a_{n -2} x^{n +r}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-2 a_{n} x^{n +r}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )-2 a_{n} x^{n +r} = 0 \] When \(n = 0\) the above becomes \[ x^{r} a_{0} r \left (-1+r \right )-2 a_{0} x^{r} = 0 \] Or \[ \left (x^{r} r \left (-1+r \right )-2 x^{r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (r^{2}-r -2\right ) x^{r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ r^{2}-r -2 = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= 2\\ r_2 &= -1 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \left (r^{2}-r -2\right ) x^{r} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \([2, -1]\).
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^{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} \end {align*}
Or \begin {align*} y_{1}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +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 -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 \(a_{0} = 1\). Substituting \(n = 1\) in Eq. (2B) gives \[ a_{1} = \frac {r}{r^{2}+r -2} \] For \(2\le n\) the recursive equation is \begin{equation} \tag{3} a_{n} \left (n +r \right ) \left (n +r -1\right )-a_{n -1} \left (n +r -1\right )+a_{n -2}-2 a_{n} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = \frac {n a_{n -1}+r a_{n -1}-a_{n -2}-a_{n -1}}{n^{2}+2 n r +r^{2}-n -r -2}\tag {4} \] Which for the root \(r = 2\) becomes \[ a_{n} = \frac {n a_{n -1}-a_{n -2}+a_{n -1}}{n \left (n +3\right )}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = 2\) and after as more terms are found using the above recursive equation.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )} \] Which for the root \(r = 2\) becomes \[ a_{2}={\frac {1}{20}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{20}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )} \] Which for the root \(r = 2\) becomes \[ a_{3}=-{\frac {1}{60}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{20}\) |
| \(a_{3}\) | \(\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(-{\frac {1}{60}}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {-r^{2}-3 r +2}{\left (r +5\right ) \left (r +2\right ) \left (r +3\right ) r \left (-1+r \right ) \left (r +4\right )} \] Which for the root \(r = 2\) becomes \[ a_{4}=-{\frac {1}{210}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{20}\) |
| \(a_{3}\) | \(\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(-{\frac {1}{60}}\) |
| \(a_{4}\) | \(\frac {-r^{2}-3 r +2}{\left (r +5\right ) \left (r +2\right ) \left (r +3\right ) r \left (-1+r \right ) \left (r +4\right )}\) | \(-{\frac {1}{210}}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=-\frac {4}{\left (r +6\right ) \left (r +3\right ) r \left (r +2\right ) \left (-1+r \right ) \left (r +4\right ) \left (r +5\right )} \] Which for the root \(r = 2\) becomes \[ a_{5}=-{\frac {1}{3360}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{20}\) |
| \(a_{3}\) | \(\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(-{\frac {1}{60}}\) |
| \(a_{4}\) | \(\frac {-r^{2}-3 r +2}{\left (r +5\right ) \left (r +2\right ) \left (r +3\right ) r \left (-1+r \right ) \left (r +4\right )}\) | \(-{\frac {1}{210}}\) |
| \(a_{5}\) | \(-\frac {4}{\left (r +6\right ) \left (r +3\right ) r \left (r +2\right ) \left (-1+r \right ) \left (r +4\right ) \left (r +5\right )}\) | \(-{\frac {1}{3360}}\) |
For \(n = 6\), using the above recursive equation gives \[ a_{6}=\frac {r^{2}+5 r -8}{\left (7+r \right ) \left (r +4\right ) \left (r +5\right ) \left (-1+r \right ) \left (r +2\right ) r \left (r +3\right ) \left (r +6\right )} \] Which for the root \(r = 2\) becomes \[ a_{6}={\frac {1}{20160}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{20}\) |
| \(a_{3}\) | \(\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(-{\frac {1}{60}}\) |
| \(a_{4}\) | \(\frac {-r^{2}-3 r +2}{\left (r +5\right ) \left (r +2\right ) \left (r +3\right ) r \left (-1+r \right ) \left (r +4\right )}\) | \(-{\frac {1}{210}}\) |
| \(a_{5}\) | \(-\frac {4}{\left (r +6\right ) \left (r +3\right ) r \left (r +2\right ) \left (-1+r \right ) \left (r +4\right ) \left (r +5\right )}\) | \(-{\frac {1}{3360}}\) |
| \(a_{6}\) | \(\frac {r^{2}+5 r -8}{\left (7+r \right ) \left (r +4\right ) \left (r +5\right ) \left (-1+r \right ) \left (r +2\right ) r \left (r +3\right ) \left (r +6\right )}\) | \(\frac {1}{20160}\) |
For \(n = 7\), using the above recursive equation gives \[ a_{7}=\frac {r^{2}+6 r -4}{\left (r +8\right ) \left (r +5\right ) \left (r +6\right ) \left (r +3\right ) r \left (r +2\right ) \left (-1+r \right ) \left (r +4\right ) \left (7+r \right )} \] Which for the root \(r = 2\) becomes \[ a_{7}={\frac {1}{100800}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{20}\) |
| \(a_{3}\) | \(\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(-{\frac {1}{60}}\) |
| \(a_{4}\) | \(\frac {-r^{2}-3 r +2}{\left (r +5\right ) \left (r +2\right ) \left (r +3\right ) r \left (-1+r \right ) \left (r +4\right )}\) | \(-{\frac {1}{210}}\) |
| \(a_{5}\) | \(-\frac {4}{\left (r +6\right ) \left (r +3\right ) r \left (r +2\right ) \left (-1+r \right ) \left (r +4\right ) \left (r +5\right )}\) | \(-{\frac {1}{3360}}\) |
| \(a_{6}\) | \(\frac {r^{2}+5 r -8}{\left (7+r \right ) \left (r +4\right ) \left (r +5\right ) \left (-1+r \right ) \left (r +2\right ) r \left (r +3\right ) \left (r +6\right )}\) | \(\frac {1}{20160}\) |
| \(a_{7}\) | \(\frac {r^{2}+6 r -4}{\left (r +8\right ) \left (r +5\right ) \left (r +6\right ) \left (r +3\right ) r \left (r +2\right ) \left (-1+r \right ) \left (r +4\right ) \left (7+r \right )}\) | \(\frac {1}{100800}\) |
Using the above table, then the solution \(y_{1}\left (x \right )\) is \begin {align*} y_{1}\left (x \right )&= x^{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}+a_{7} x^{7}+a_{8} x^{8}\dots \right ) \\ &= x^{2} \left (1+\frac {x}{2}+\frac {x^{2}}{20}-\frac {x^{3}}{60}-\frac {x^{4}}{210}-\frac {x^{5}}{3360}+\frac {x^{6}}{20160}+\frac {x^{7}}{100800}+O\left (x^{8}\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 {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )} \end {align*}
Therefore \begin {align*} \lim _{r\rightarrow r_{2}}\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}&= \lim _{r\rightarrow -1}\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}\\ &= {\frac {5}{12}} \end {align*}
The limit is \(\frac {5}{12}\). Since the limit exists then the log term is not needed and we can set \(C = 0\). Therefore the second solution has the form \begin {align*} y_{2}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n +r}\\ &= \moverset {\infty }{\munderset {n =0}{\sum }}b_{n} x^{n -1} \end {align*}
Eq (3) derived above is used to find all \(b_{n}\) coefficients. The case \(n = 0\) is skipped since it was used to find the roots of the indicial equation. \(b_{0}\) is arbitrary and taken as \(b_{0} = 1\). Substituting \(n = 1\) in Eq(3) gives \[ b_{1} = \frac {r}{r^{2}+r -2} \] For \(2\le n\) the recursive equation is \begin{equation} \tag{4} b_{n} \left (n +r \right ) \left (n +r -1\right )-b_{n -1} \left (n +r -1\right )+b_{n -2}-2 b_{n} = 0 \end{equation} Which for for the root \(r = -1\) becomes \begin{equation} \tag{4A} b_{n} \left (n -1\right ) \left (n -2\right )-b_{n -1} \left (n -2\right )+b_{n -2}-2 b_{n} = 0 \end{equation} Solving for \(b_{n}\) from the recursive equation (4) gives \[ b_{n} = \frac {n b_{n -1}+r b_{n -1}-b_{n -2}-b_{n -1}}{n^{2}+2 n r +r^{2}-n -r -2}\tag {5} \] Which for the root \(r = -1\) becomes \[ b_{n} = \frac {n b_{n -1}-b_{n -2}-2 b_{n -1}}{n^{2}-3 n}\tag {6} \] At this point, it is a good idea to keep track of \(b_{n}\) in a table both before substituting \(r = -1\) and after as more terms are found using the above recursive equation.
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )} \] Which for the root \(r = -1\) becomes \[ b_{2}={\frac {1}{2}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(b_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{2}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=-\frac {r^{2}+2 r -4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )} \] Which for the root \(r = -1\) becomes \[ b_{3}={\frac {5}{12}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(b_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {5}{12}\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=-\frac {r^{2}+3 r -2}{\left (r^{2}+7 r +10\right ) \left (r +3\right ) r \left (-1+r \right ) \left (r +4\right )} \] Which for the root \(r = -1\) becomes \[ b_{4}={\frac {1}{12}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(b_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {5}{12}\) |
| \(b_{4}\) | \(\frac {-r^{2}-3 r +2}{\left (r +5\right ) \left (r +2\right ) \left (r +3\right ) r \left (-1+r \right ) \left (r +4\right )}\) | \(\frac {1}{12}\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=-\frac {4}{\left (r^{2}+9 r +18\right ) r \left (r^{2}+r -2\right ) \left (r +4\right ) \left (r +5\right )} \] Which for the root \(r = -1\) becomes \[ b_{5}=-{\frac {1}{60}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(b_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {5}{12}\) |
| \(b_{4}\) | \(\frac {-r^{2}-3 r +2}{\left (r +5\right ) \left (r +2\right ) \left (r +3\right ) r \left (-1+r \right ) \left (r +4\right )}\) | \(\frac {1}{12}\) |
| \(b_{5}\) | \(-\frac {4}{\left (r +6\right ) \left (r +3\right ) r \left (r +2\right ) \left (-1+r \right ) \left (r +4\right ) \left (r +5\right )}\) | \(-{\frac {1}{60}}\) |
For \(n = 6\), using the above recursive equation gives \[ b_{6}=\frac {r^{2}+5 r -8}{\left (-1+r \right ) r \left (r +3\right ) \left (r^{2}+7 r +10\right ) \left (r +6\right ) \left (r^{2}+11 r +28\right )} \] Which for the root \(r = -1\) becomes \[ b_{6}=-{\frac {1}{120}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(b_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {5}{12}\) |
| \(b_{4}\) | \(\frac {-r^{2}-3 r +2}{\left (r +5\right ) \left (r +2\right ) \left (r +3\right ) r \left (-1+r \right ) \left (r +4\right )}\) | \(\frac {1}{12}\) |
| \(b_{5}\) | \(-\frac {4}{\left (r +6\right ) \left (r +3\right ) r \left (r +2\right ) \left (-1+r \right ) \left (r +4\right ) \left (r +5\right )}\) | \(-{\frac {1}{60}}\) |
| \(b_{6}\) | \(\frac {r^{2}+5 r -8}{\left (7+r \right ) \left (r +4\right ) \left (r +5\right ) \left (-1+r \right ) \left (r +2\right ) r \left (r +3\right ) \left (r +6\right )}\) | \(-{\frac {1}{120}}\) |
For \(n = 7\), using the above recursive equation gives \[ b_{7}=\frac {r^{2}+6 r -4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r^{2}+9 r +18\right ) \left (7+r \right ) \left (r^{2}+13 r +40\right )} \] Which for the root \(r = -1\) becomes \[ b_{7}=-{\frac {1}{1120}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r}{r^{2}+r -2}\) | \(\frac {1}{2}\) |
| \(b_{2}\) | \(\frac {2}{\left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {1}{2}\) |
| \(b_{3}\) | \(\frac {-r^{2}-2 r +4}{\left (r +4\right ) \left (r^{2}+r -2\right ) r \left (r +3\right )}\) | \(\frac {5}{12}\) |
| \(b_{4}\) | \(\frac {-r^{2}-3 r +2}{\left (r +5\right ) \left (r +2\right ) \left (r +3\right ) r \left (-1+r \right ) \left (r +4\right )}\) | \(\frac {1}{12}\) |
| \(b_{5}\) | \(-\frac {4}{\left (r +6\right ) \left (r +3\right ) r \left (r +2\right ) \left (-1+r \right ) \left (r +4\right ) \left (r +5\right )}\) | \(-{\frac {1}{60}}\) |
| \(b_{6}\) | \(\frac {r^{2}+5 r -8}{\left (7+r \right ) \left (r +4\right ) \left (r +5\right ) \left (-1+r \right ) \left (r +2\right ) r \left (r +3\right ) \left (r +6\right )}\) | \(-{\frac {1}{120}}\) |
| \(b_{7}\) | \(\frac {r^{2}+6 r -4}{\left (r +8\right ) \left (r +5\right ) \left (r +6\right ) \left (r +3\right ) r \left (r +2\right ) \left (-1+r \right ) \left (r +4\right ) \left (7+r \right )}\) | \(-{\frac {1}{1120}}\) |
Using the above table, then the solution \(y_{2}\left (x \right )\) is \begin {align*} y_{2}\left (x \right )&= x^{2} \left (b_{0}+b_{1} x +b_{2} x^{2}+b_{3} x^{3}+b_{4} x^{4}+b_{5} x^{5}+b_{6} x^{6}+b_{7} x^{7}+b_{8} x^{8}\dots \right ) \\ &= \frac {1+\frac {x}{2}+\frac {x^{2}}{2}+\frac {5 x^{3}}{12}+\frac {x^{4}}{12}-\frac {x^{5}}{60}-\frac {x^{6}}{120}-\frac {x^{7}}{1120}+O\left (x^{8}\right )}{x} \end {align*}
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^{2} \left (1+\frac {x}{2}+\frac {x^{2}}{20}-\frac {x^{3}}{60}-\frac {x^{4}}{210}-\frac {x^{5}}{3360}+\frac {x^{6}}{20160}+\frac {x^{7}}{100800}+O\left (x^{8}\right )\right ) + \frac {c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{2}+\frac {5 x^{3}}{12}+\frac {x^{4}}{12}-\frac {x^{5}}{60}-\frac {x^{6}}{120}-\frac {x^{7}}{1120}+O\left (x^{8}\right )\right )}{x} \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} x^{2} \left (1+\frac {x}{2}+\frac {x^{2}}{20}-\frac {x^{3}}{60}-\frac {x^{4}}{210}-\frac {x^{5}}{3360}+\frac {x^{6}}{20160}+\frac {x^{7}}{100800}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{2}+\frac {5 x^{3}}{12}+\frac {x^{4}}{12}-\frac {x^{5}}{60}-\frac {x^{6}}{120}-\frac {x^{7}}{1120}+O\left (x^{8}\right )\right )}{x} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{2} \left (1+\frac {x}{2}+\frac {x^{2}}{20}-\frac {x^{3}}{60}-\frac {x^{4}}{210}-\frac {x^{5}}{3360}+\frac {x^{6}}{20160}+\frac {x^{7}}{100800}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{2}+\frac {5 x^{3}}{12}+\frac {x^{4}}{12}-\frac {x^{5}}{60}-\frac {x^{6}}{120}-\frac {x^{7}}{1120}+O\left (x^{8}\right )\right )}{x} \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{2} \left (1+\frac {x}{2}+\frac {x^{2}}{20}-\frac {x^{3}}{60}-\frac {x^{4}}{210}-\frac {x^{5}}{3360}+\frac {x^{6}}{20160}+\frac {x^{7}}{100800}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{2}+\frac {5 x^{3}}{12}+\frac {x^{4}}{12}-\frac {x^{5}}{60}-\frac {x^{6}}{120}-\frac {x^{7}}{1120}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x^{2}-2\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=y^{\prime }-\frac {\left (x^{2}-2\right ) y}{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} \\ {} & {} & y^{\prime \prime }-y^{\prime }+\frac {\left (x^{2}-2\right ) y}{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 )=-1, P_{3}\left (x \right )=\frac {x^{2}-2}{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}}}=0 \\ {} & \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}}}=-2 \\ {} & \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} \\ {} & {} & x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x^{2}-2\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..2 \\ {} & {} & 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^{2}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot 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 \\ {} & {} & x^{2}\cdot 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^{2}\cdot y^{\prime \prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot y^{\prime \prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k -1+r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (1+r \right ) \left (-2+r \right ) x^{r}+\left (a_{1} \left (2+r \right ) \left (-1+r \right )-a_{0} r \right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (k +r +1\right ) \left (k +r -2\right )-a_{k -1} \left (k -1+r \right )+a_{k -2}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \left (1+r \right ) \left (-2+r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-1, 2\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & a_{1} \left (2+r \right ) \left (-1+r \right )-a_{0} r =0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=\frac {a_{0} r}{r^{2}+r -2} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & a_{k} \left (k +r +1\right ) \left (k +r -2\right )-a_{k -1} k -a_{k -1} r +a_{k -2}+a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & a_{k +2} \left (k +3+r \right ) \left (k +r \right )-a_{k +1} \left (k +2\right )-a_{k +1} r +a_{k}+a_{k +1}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=\frac {k a_{k +1}+a_{k +1} r -a_{k}+a_{k +1}}{\left (k +3+r \right ) \left (k +r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-1 \\ {} & {} & a_{k +2}=\frac {k a_{k +1}-a_{k}}{\left (k +2\right ) \left (k -1\right )} \\ \bullet & {} & \textrm {Series not valid for}\hspace {3pt} r =-1\hspace {3pt}\textrm {, division by}\hspace {3pt} 0\hspace {3pt}\textrm {in the recursion relation at}\hspace {3pt} k =1 \\ {} & {} & a_{k +2}=\frac {k a_{k +1}-a_{k}}{\left (k +2\right ) \left (k -1\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =2 \\ {} & {} & a_{k +2}=\frac {k a_{k +1}-a_{k}+3 a_{k +1}}{\left (k +5\right ) \left (k +2\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =2 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +2}, a_{k +2}=\frac {k a_{k +1}-a_{k}+3 a_{k +1}}{\left (k +5\right ) \left (k +2\right )}, a_{1}=\frac {a_{0}}{2}\right ] \end {array} \]
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 Group is reducible or imprimitive <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 55
Order:=8; dsolve(x^2*diff(y(x),x$2)-x^2*diff(y(x),x)+(x^2-2)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{2} \left (1+\frac {1}{2} x +\frac {1}{20} x^{2}-\frac {1}{60} x^{3}-\frac {1}{210} x^{4}-\frac {1}{3360} x^{5}+\frac {1}{20160} x^{6}+\frac {1}{100800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12+6 x +6 x^{2}+5 x^{3}+x^{4}-\frac {1}{5} x^{5}-\frac {1}{10} x^{6}-\frac {3}{280} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.069 (sec). Leaf size: 96
AsymptoticDSolveValue[x^2*y''[x]-x^2*y'[x]+(x^2-2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {x^5}{120}-\frac {x^4}{60}+\frac {x^3}{12}+\frac {5 x^2}{12}+\frac {x}{2}+\frac {1}{x}+\frac {1}{2}\right )+c_2 \left (\frac {x^8}{20160}-\frac {x^7}{3360}-\frac {x^6}{210}-\frac {x^5}{60}+\frac {x^4}{20}+\frac {x^3}{2}+x^2\right ) \]