Internal problem ID [11903]
Internal file name [OUTPUT/11913_Saturday_April_13_2024_10_26_15_PM_27176372/index.tex]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius).
Exercises page 251
Problem number: 3.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Irregular singular point"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {\left (x^{4}-2 x^{3}+x^{2}\right ) y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }+x^{2} y=0} \] 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. \[ \left (x^{4}-2 x^{3}+x^{2}\right ) y^{\prime \prime }+\left (2 x -2\right ) y^{\prime }+x^{2} y = 0 \] 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}{\left (x -1\right ) x^{2}}\\ q(x) &= \frac {1}{\left (x -1\right )^{2}}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([1, \infty ]\)
Irregular singular points : \([0]\)
Since \(x = 0\) is not an ordinary point, then we will now check if it is a regular singular point. Unable to solve since \(x = 0\) is not regular singular point. Terminating.
Verification of solutions N/A
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime } x^{2} \left (x -1\right )^{2}+\left (2 x -2\right ) y^{\prime }+x^{2} 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 }=-\frac {2 y^{\prime }}{\left (x -1\right ) x^{2}}-\frac {y}{\left (x -1\right )^{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 }+\frac {2 y^{\prime }}{\left (x -1\right ) x^{2}}+\frac {y}{\left (x -1\right )^{2}}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {2}{\left (x -1\right ) x^{2}}, P_{3}\left (x \right )=\frac {1}{\left (x -1\right )^{2}}\right ] \\ {} & \circ & \left (x -1\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =1 \\ {} & {} & \left (\left (x -1\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}1}}}=2 \\ {} & \circ & \left (x -1\right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =1 \\ {} & {} & \left (\left (x -1\right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}1}}}=1 \\ {} & \circ & x =1\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=1 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & y^{\prime \prime } x^{2} \left (x -1\right )^{2}+\left (2 x -2\right ) y^{\prime }+x^{2} y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u +1\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (u^{4}+2 u^{3}+u^{2}\right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+2 u \left (\frac {d}{d u}y \left (u \right )\right )+\left (u^{2}+2 u +1\right ) y \left (u \right )=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \left (u \right ) \\ {} & {} & y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot y \left (u \right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u \cdot \left (\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & u \cdot \left (\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =2..4 \\ {} & {} & u^{m}\cdot \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) u^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & u^{m}\cdot \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) u^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (r^{2}+r +1\right ) u^{r}+\left (\left (r^{2}+3 r +3\right ) a_{1}+2 a_{0} \left (r^{2}-r +1\right )\right ) u^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (k^{2}+2 k r +r^{2}+k +r +1\right )+2 a_{k -1} \left (\left (k -1\right )^{2}+2 \left (k -1\right ) r +r^{2}-k +2-r \right )+a_{k -2} \left (\left (k -2\right )^{2}+2 \left (k -2\right ) r +r^{2}-k +3-r \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & r^{2}+r +1=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}, -\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & \left (r^{2}+3 r +3\right ) a_{1}+2 a_{0} \left (r^{2}-r +1\right )=0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=-\frac {2 a_{0} \left (r^{2}-r +1\right )}{r^{2}+3 r +3} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (a_{k}+a_{k -2}+2 a_{k -1}\right ) k^{2}+\left (\left (2 a_{k}+2 a_{k -2}+4 a_{k -1}\right ) r +a_{k}-5 a_{k -2}-6 a_{k -1}\right ) k +\left (a_{k}+a_{k -2}+2 a_{k -1}\right ) r^{2}+\left (a_{k}-5 a_{k -2}-6 a_{k -1}\right ) r +a_{k}+7 a_{k -2}+6 a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \left (a_{k +2}+a_{k}+2 a_{k +1}\right ) \left (k +2\right )^{2}+\left (\left (2 a_{k +2}+2 a_{k}+4 a_{k +1}\right ) r +a_{k +2}-5 a_{k}-6 a_{k +1}\right ) \left (k +2\right )+\left (a_{k +2}+a_{k}+2 a_{k +1}\right ) r^{2}+\left (a_{k +2}-5 a_{k}-6 a_{k +1}\right ) r +a_{k +2}+7 a_{k}+6 a_{k +1}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {k^{2} a_{k}+2 k^{2} a_{k +1}+2 k r a_{k}+4 k r a_{k +1}+r^{2} a_{k}+2 r^{2} a_{k +1}-k a_{k}+2 k a_{k +1}-r a_{k}+2 r a_{k +1}+a_{k}+2 a_{k +1}}{k^{2}+2 k r +r^{2}+5 k +5 r +7} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2} \\ {} & {} & a_{k +2}=-\frac {k^{2} a_{k}+2 k^{2} a_{k +1}+2 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+4 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k}+2 \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k +1}-k a_{k}+2 k a_{k +1}-\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+2 \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+a_{k}+2 a_{k +1}}{k^{2}+2 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )+\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+5 k +\frac {9}{2}-\frac {5 \,\mathrm {I} \sqrt {3}}{2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k -\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}}, a_{k +2}=-\frac {k^{2} a_{k}+2 k^{2} a_{k +1}+2 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+4 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k}+2 \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k +1}-k a_{k}+2 k a_{k +1}-\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+2 \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+a_{k}+2 a_{k +1}}{k^{2}+2 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )+\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+5 k +\frac {9}{2}-\frac {5 \,\mathrm {I} \sqrt {3}}{2}}, a_{1}=-\frac {2 a_{0} \left (\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x -1\right )^{k -\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}}, a_{k +2}=-\frac {k^{2} a_{k}+2 k^{2} a_{k +1}+2 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+4 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k}+2 \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k +1}-k a_{k}+2 k a_{k +1}-\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+2 \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+a_{k}+2 a_{k +1}}{k^{2}+2 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )+\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+5 k +\frac {9}{2}-\frac {5 \,\mathrm {I} \sqrt {3}}{2}}, a_{1}=-\frac {2 a_{0} \left (\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2} \\ {} & {} & a_{k +2}=-\frac {k^{2} a_{k}+2 k^{2} a_{k +1}+2 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+4 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k}+2 \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k +1}-k a_{k}+2 k a_{k +1}-\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+2 \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+a_{k}+2 a_{k +1}}{k^{2}+2 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )+\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+5 k +\frac {9}{2}+\frac {5 \,\mathrm {I} \sqrt {3}}{2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k -\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}}, a_{k +2}=-\frac {k^{2} a_{k}+2 k^{2} a_{k +1}+2 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+4 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k}+2 \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k +1}-k a_{k}+2 k a_{k +1}-\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+2 \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+a_{k}+2 a_{k +1}}{k^{2}+2 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )+\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+5 k +\frac {9}{2}+\frac {5 \,\mathrm {I} \sqrt {3}}{2}}, a_{1}=-\frac {2 a_{0} \left (\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}+\frac {3 \,\mathrm {I} \sqrt {3}}{2}}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x -1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x -1\right )^{k -\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}}, a_{k +2}=-\frac {k^{2} a_{k}+2 k^{2} a_{k +1}+2 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+4 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k}+2 \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k +1}-k a_{k}+2 k a_{k +1}-\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+2 \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+a_{k}+2 a_{k +1}}{k^{2}+2 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )+\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+5 k +\frac {9}{2}+\frac {5 \,\mathrm {I} \sqrt {3}}{2}}, a_{1}=-\frac {2 a_{0} \left (\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}+\frac {3 \,\mathrm {I} \sqrt {3}}{2}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x -1\right )^{k -\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} \left (x -1\right )^{k -\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}}\right ), a_{2+k}=-\frac {k^{2} a_{k}+2 k^{2} a_{k +1}+2 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+4 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k}+2 \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} a_{k +1}-k a_{k}+2 k a_{k +1}-\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k}+2 \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) a_{k +1}+a_{k}+2 a_{k +1}}{k^{2}+2 k \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )+\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+5 k +\frac {9}{2}-\frac {5 \,\mathrm {I} \sqrt {3}}{2}}, a_{1}=-\frac {2 a_{0} \left (\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}-\frac {3 \,\mathrm {I} \sqrt {3}}{2}}, b_{2+k}=-\frac {k^{2} b_{k}+2 k^{2} b_{k +1}+2 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) b_{k}+4 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) b_{k +1}+\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} b_{k}+2 \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2} b_{k +1}-k b_{k}+2 k b_{k +1}-\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) b_{k}+2 \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) b_{k +1}+b_{k}+2 b_{k +1}}{k^{2}+2 k \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )+\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+5 k +\frac {9}{2}+\frac {5 \,\mathrm {I} \sqrt {3}}{2}}, b_{1}=-\frac {2 b_{0} \left (\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}+\frac {3}{2}+\frac {3 \,\mathrm {I} \sqrt {3}}{2}}\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 -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunC ODE, case a = 0, e <> 0, c <> 0 `
✗ Solution by Maple
Order:=6; dsolve((x^4-2*x^3+x^2)*diff(y(x),x$2)+2*(x-1)*diff(y(x),x)+x^2*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.047 (sec). Leaf size: 71
AsymptoticDSolveValue[(x^4-2*x^3+x^2)*y''[x]+2*(x-1)*y'[x]+x^2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {3 x^5}{10}+\frac {x^4}{4}+\frac {x^3}{6}+1\right )+c_2 e^{-2/x} \left (-\frac {429 x^5}{5}+\frac {91 x^4}{4}-\frac {31 x^3}{6}+3 x^2+1\right ) x^4 \]