Internal problem ID [5564]
Internal file name [OUTPUT/4812_Sunday_June_05_2022_03_06_31_PM_60913564/index.tex]
Book: A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications.
Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section: Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.2 page
239
Problem number: 9.
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.
\[ \boxed {x^{3} \left (x^{2}-25\right ) \left (x -2\right )^{2} y^{\prime \prime }+3 x \left (x -2\right ) y^{\prime }+7 \left (x +5\right ) y=0} \] 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. \[ \left (x^{7}-4 x^{6}-21 x^{5}+100 x^{4}-100 x^{3}\right ) y^{\prime \prime }+\left (3 x^{2}-6 x \right ) y^{\prime }+\left (7 x +35\right ) y = 0 \] 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 {3}{x^{2} \left (x -2\right ) \left (x -5\right ) \left (x +5\right )}\\ q(x) &= \frac {7}{x^{3} \left (x -5\right ) \left (x -2\right )^{2}}\\ \end {align*}
| \(p(x)=\frac {3}{x^{2} \left (x -2\right ) \left (x -5\right ) \left (x +5\right )}\) | |
| singularity | type |
| \(x = -5\) | \(\text {``regular''}\) |
| \(x = 0\) | \(\text {``irregular''}\) |
| \(x = 2\) | \(\text {``regular''}\) |
| \(x = 5\) | \(\text {``regular''}\) |
| \(q(x)=\frac {7}{x^{3} \left (x -5\right ) \left (x -2\right )^{2}}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``irregular''}\) |
| \(x = 2\) | \(\text {``regular''}\) |
| \(x = 5\) | \(\text {``regular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([-5, 2, 5, \infty ]\)
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^{3} \left (x -5\right ) \left (x +5\right ) \left (x -2\right )^{2}+\left (3 x^{2}-6 x \right ) y^{\prime }+\left (7 x +35\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 }=-\frac {3 y^{\prime }}{x^{2} \left (x -2\right ) \left (x -5\right ) \left (x +5\right )}-\frac {7 y}{x^{3} \left (x -5\right ) \left (x -2\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 {3 y^{\prime }}{x^{2} \left (x -2\right ) \left (x -5\right ) \left (x +5\right )}+\frac {7 y}{x^{3} \left (x -5\right ) \left (x -2\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 {3}{x^{2} \left (x -2\right ) \left (x -5\right ) \left (x +5\right )}, P_{3}\left (x \right )=\frac {7}{x^{3} \left (x -5\right ) \left (x -2\right )^{2}}\right ] \\ {} & \circ & \left (x +5\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-5 \\ {} & {} & \left (\left (x +5\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-5}}}=\frac {3}{1750} \\ {} & \circ & \left (x +5\right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-5 \\ {} & {} & \left (\left (x +5\right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-5}}}=0 \\ {} & \circ & x =-5\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}=-5 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & y^{\prime \prime } x^{3} \left (x -5\right ) \left (x +5\right ) \left (x -2\right )^{2}+3 x \left (x -2\right ) y^{\prime }+\left (7 x +35\right ) y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u -5\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & \left (u^{7}-39 u^{6}+624 u^{5}-5250 u^{4}+24525 u^{3}-60375 u^{2}+61250 u \right ) \left (\frac {d^{2}}{d u^{2}}y \left (u \right )\right )+\left (3 u^{2}-36 u +105\right ) \left (\frac {d}{d u}y \left (u \right )\right )+7 u 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 \cdot y \left (u \right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & u \cdot y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r +1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -1 \\ {} & {} & u \cdot y \left (u \right )=\moverset {\infty }{\munderset {k =1}{\sum }}a_{k -1} u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & u^{m}\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 -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +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 =1..7 \\ {} & {} & 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} \\ {} & {} & 35 a_{0} r \left (-1747+1750 r \right ) u^{-1+r}+\left (35 a_{1} \left (1+r \right ) \left (3+1750 r \right )-3 a_{0} r \left (-20113+20125 r \right )\right ) u^{r}+\left (35 a_{2} \left (2+r \right ) \left (1753+1750 r \right )-3 a_{1} \left (1+r \right ) \left (12+20125 r \right )+a_{0} \left (24525 r^{2}-24522 r +7\right )\right ) u^{1+r}+\left (35 a_{3} \left (3+r \right ) \left (3503+1750 r \right )-3 a_{2} \left (2+r \right ) \left (20137+20125 r \right )+a_{1} \left (24525 r^{2}+24528 r +10\right )-5250 a_{0} r \left (-1+r \right )\right ) u^{2+r}+\left (35 a_{4} \left (4+r \right ) \left (5253+1750 r \right )-3 a_{3} \left (3+r \right ) \left (40262+20125 r \right )+a_{2} \left (24525 r^{2}+73578 r +49063\right )-5250 a_{1} \left (1+r \right ) r +624 a_{0} r \left (-1+r \right )\right ) u^{3+r}+\left (35 a_{5} \left (5+r \right ) \left (7003+1750 r \right )-3 a_{4} \left (4+r \right ) \left (60387+20125 r \right )+a_{3} \left (24525 r^{2}+122628 r +147166\right )-5250 a_{2} \left (2+r \right ) \left (1+r \right )+624 a_{1} \left (1+r \right ) r -39 a_{0} r \left (-1+r \right )\right ) u^{4+r}+\left (\moverset {\infty }{\munderset {k =5}{\sum }}\left (35 a_{k +1} \left (k +r +1\right ) \left (1750 k +3+1750 r \right )-3 a_{k} \left (k +r \right ) \left (20125 k +20125 r -20113\right )+a_{k -1} \left (24525 \left (k -1\right )^{2}+49050 \left (k -1\right ) r +24525 r^{2}-24522 k +24529-24522 r \right )-5250 a_{k -2} \left (k -2+r \right ) \left (k -3+r \right )+624 a_{k -3} \left (k -3+r \right ) \left (k -4+r \right )-39 a_{k -4} \left (k -4+r \right ) \left (k -5+r \right )+a_{k -5} \left (k -5+r \right ) \left (k -6+r \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & 35 r \left (-1747+1750 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, \frac {1747}{1750}\right \} \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} u \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [35 a_{1} \left (1+r \right ) \left (3+1750 r \right )-3 a_{0} r \left (-20113+20125 r \right )=0, 35 a_{2} \left (2+r \right ) \left (1753+1750 r \right )-3 a_{1} \left (1+r \right ) \left (12+20125 r \right )+a_{0} \left (24525 r^{2}-24522 r +7\right )=0, 35 a_{3} \left (3+r \right ) \left (3503+1750 r \right )-3 a_{2} \left (2+r \right ) \left (20137+20125 r \right )+a_{1} \left (24525 r^{2}+24528 r +10\right )-5250 a_{0} r \left (-1+r \right )=0, 35 a_{4} \left (4+r \right ) \left (5253+1750 r \right )-3 a_{3} \left (3+r \right ) \left (40262+20125 r \right )+a_{2} \left (24525 r^{2}+73578 r +49063\right )-5250 a_{1} \left (1+r \right ) r +624 a_{0} r \left (-1+r \right )=0, 35 a_{5} \left (5+r \right ) \left (7003+1750 r \right )-3 a_{4} \left (4+r \right ) \left (60387+20125 r \right )+a_{3} \left (24525 r^{2}+122628 r +147166\right )-5250 a_{2} \left (2+r \right ) \left (1+r \right )+624 a_{1} \left (1+r \right ) r -39 a_{0} r \left (-1+r \right )=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{1}=\frac {3 a_{0} r \left (-20113+20125 r \right )}{35 \left (1750 r^{2}+1753 r +3\right )}, a_{2}=\frac {a_{0} \left (2142984375 r^{3}-2141396250 r^{2}-26144 r -735\right )}{1225 \left (3062500 r^{3}+9198000 r^{2}+6151259 r +10518\right )}, a_{3}=\frac {3 a_{0} \left (19461900390625 r^{5}+19468538359375 r^{4}-19435629227000 r^{3}-19441771674383 r^{2}+510295602 r -14800695\right )}{42875 \left (5359375000 r^{5}+37543187500 r^{4}+91274797250 r^{3}+91431173777 r^{2}+32395455858 r +55266831\right )}, a_{4}=\frac {a_{0} \left (1351682693115234375 r^{7}+6757468463730468750 r^{6}+9459902927605265625 r^{5}-1346741085129720000 r^{4}-10798918506236535254 r^{3}-5396543958863751642 r^{2}+1515360202142957 r -6305004178095\right )}{1500625 \left (9378906250000 r^{7}+121990093750000 r^{6}+629126477875000 r^{5}+1644590185189000 r^{4}+2295597905419081 r^{3}+1621124864443567 r^{2}+454376031607134 r +774177768648\right )}, a_{5}=\frac {3 a_{0} \left (9334340984771728515625 r^{9}+102654645979007568359375 r^{8}+429169625497935943359375 r^{7}+802090179309157053875000 r^{6}+456715295150926645105125 r^{5}-568733232395119777617403 r^{4}-894216179388590294105623 r^{3}-334778918752075306271568 r^{2}+229330342344661329417 r -515254338972328095\right )}{52521875 \left (16413085937500000 r^{9}+344815488281250000 r^{8}+3055507455000000000 r^{7}+14891366599108125000 r^{6}+43568358836231958750 r^{5}+78171579739984526493 r^{4}+83781322696910591423 r^{3}+48936832887274798713 r^{2}+11943323516495557323 r +20330875926907290\right )}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (-60375 a_{k}+a_{k -5}-39 a_{k -4}+624 a_{k -3}-5250 a_{k -2}+24525 a_{k -1}+61250 a_{k +1}\right ) k^{2}+\left (2 \left (-60375 a_{k}+a_{k -5}-39 a_{k -4}+624 a_{k -3}-5250 a_{k -2}+24525 a_{k -1}+61250 a_{k +1}\right ) r +60339 a_{k}-11 a_{k -5}+351 a_{k -4}-4368 a_{k -3}+26250 a_{k -2}-73572 a_{k -1}+61355 a_{k +1}\right ) k +\left (-60375 a_{k}+a_{k -5}-39 a_{k -4}+624 a_{k -3}-5250 a_{k -2}+24525 a_{k -1}+61250 a_{k +1}\right ) r^{2}+\left (60339 a_{k}-11 a_{k -5}+351 a_{k -4}-4368 a_{k -3}+26250 a_{k -2}-73572 a_{k -1}+61355 a_{k +1}\right ) r +30 a_{k -5}-780 a_{k -4}+7488 a_{k -3}-31500 a_{k -2}+49054 a_{k -1}+105 a_{k +1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +5 \\ {} & {} & \left (-60375 a_{k +5}+a_{k}-39 a_{k +1}+624 a_{k +2}-5250 a_{k +3}+24525 a_{k +4}+61250 a_{k +6}\right ) \left (k +5\right )^{2}+\left (2 \left (-60375 a_{k +5}+a_{k}-39 a_{k +1}+624 a_{k +2}-5250 a_{k +3}+24525 a_{k +4}+61250 a_{k +6}\right ) r +60339 a_{k +5}-11 a_{k}+351 a_{k +1}-4368 a_{k +2}+26250 a_{k +3}-73572 a_{k +4}+61355 a_{k +6}\right ) \left (k +5\right )+\left (-60375 a_{k +5}+a_{k}-39 a_{k +1}+624 a_{k +2}-5250 a_{k +3}+24525 a_{k +4}+61250 a_{k +6}\right ) r^{2}+\left (60339 a_{k +5}-11 a_{k}+351 a_{k +1}-4368 a_{k +2}+26250 a_{k +3}-73572 a_{k +4}+61355 a_{k +6}\right ) r +30 a_{k}-780 a_{k +1}+7488 a_{k +2}-31500 a_{k +3}+49054 a_{k +4}+105 a_{k +6}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +6}=-\frac {k^{2} a_{k}-39 k^{2} a_{k +1}+624 k^{2} a_{k +2}-5250 k^{2} a_{k +3}+24525 k^{2} a_{k +4}-60375 k^{2} a_{k +5}+2 k r a_{k}-78 k r a_{k +1}+1248 k r a_{k +2}-10500 k r a_{k +3}+49050 k r a_{k +4}-120750 k r a_{k +5}+r^{2} a_{k}-39 r^{2} a_{k +1}+624 r^{2} a_{k +2}-5250 r^{2} a_{k +3}+24525 r^{2} a_{k +4}-60375 r^{2} a_{k +5}-k a_{k}-39 k a_{k +1}+1872 k a_{k +2}-26250 k a_{k +3}+171678 k a_{k +4}-543411 k a_{k +5}-r a_{k}-39 r a_{k +1}+1872 r a_{k +2}-26250 r a_{k +3}+171678 r a_{k +4}-543411 r a_{k +5}+1248 a_{k +2}-31500 a_{k +3}+294319 a_{k +4}-1207680 a_{k +5}}{35 \left (1750 k^{2}+3500 k r +1750 r^{2}+19253 k +19253 r +52518\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +6}=-\frac {k^{2} a_{k}-39 k^{2} a_{k +1}+624 k^{2} a_{k +2}-5250 k^{2} a_{k +3}+24525 k^{2} a_{k +4}-60375 k^{2} a_{k +5}-k a_{k}-39 k a_{k +1}+1872 k a_{k +2}-26250 k a_{k +3}+171678 k a_{k +4}-543411 k a_{k +5}+1248 a_{k +2}-31500 a_{k +3}+294319 a_{k +4}-1207680 a_{k +5}}{35 \left (1750 k^{2}+19253 k +52518\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k}, a_{k +6}=-\frac {k^{2} a_{k}-39 k^{2} a_{k +1}+624 k^{2} a_{k +2}-5250 k^{2} a_{k +3}+24525 k^{2} a_{k +4}-60375 k^{2} a_{k +5}-k a_{k}-39 k a_{k +1}+1872 k a_{k +2}-26250 k a_{k +3}+171678 k a_{k +4}-543411 k a_{k +5}+1248 a_{k +2}-31500 a_{k +3}+294319 a_{k +4}-1207680 a_{k +5}}{35 \left (1750 k^{2}+19253 k +52518\right )}, a_{1}=0, a_{2}=-\frac {a_{0}}{17530}, a_{3}=-\frac {20137 a_{0}}{1074632825}, a_{4}=-\frac {8578236977 a_{0}}{1580612944323000}, a_{5}=-\frac {233675437175659 a_{0}}{161423389882620381250}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +5 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +5\right )^{k}, a_{k +6}=-\frac {k^{2} a_{k}-39 k^{2} a_{k +1}+624 k^{2} a_{k +2}-5250 k^{2} a_{k +3}+24525 k^{2} a_{k +4}-60375 k^{2} a_{k +5}-k a_{k}-39 k a_{k +1}+1872 k a_{k +2}-26250 k a_{k +3}+171678 k a_{k +4}-543411 k a_{k +5}+1248 a_{k +2}-31500 a_{k +3}+294319 a_{k +4}-1207680 a_{k +5}}{35 \left (1750 k^{2}+19253 k +52518\right )}, a_{1}=0, a_{2}=-\frac {a_{0}}{17530}, a_{3}=-\frac {20137 a_{0}}{1074632825}, a_{4}=-\frac {8578236977 a_{0}}{1580612944323000}, a_{5}=-\frac {233675437175659 a_{0}}{161423389882620381250}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1747}{1750} \\ {} & {} & a_{k +6}=-\frac {k^{2} a_{k}-39 k^{2} a_{k +1}+624 k^{2} a_{k +2}-5250 k^{2} a_{k +3}+24525 k^{2} a_{k +4}-60375 k^{2} a_{k +5}+\frac {872}{875} k a_{k}-\frac {102258}{875} k a_{k +1}+\frac {2728128}{875} k a_{k +2}-36732 k a_{k +3}+\frac {7722537}{35} k a_{k +4}-663954 k a_{k +5}-\frac {5241}{3062500} a_{k}-\frac {238261101}{3062500} a_{k +1}+\frac {2862406404}{765625} a_{k +2}-\frac {110139777}{1750} a_{k +3}+\frac {60042600949}{122500} a_{k +4}-\frac {1267229331}{700} a_{k +5}}{35 \left (1750 k^{2}+22747 k +73482\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {1747}{1750} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {1747}{1750}}, a_{k +6}=-\frac {k^{2} a_{k}-39 k^{2} a_{k +1}+624 k^{2} a_{k +2}-5250 k^{2} a_{k +3}+24525 k^{2} a_{k +4}-60375 k^{2} a_{k +5}+\frac {872}{875} k a_{k}-\frac {102258}{875} k a_{k +1}+\frac {2728128}{875} k a_{k +2}-36732 k a_{k +3}+\frac {7722537}{35} k a_{k +4}-663954 k a_{k +5}-\frac {5241}{3062500} a_{k}-\frac {238261101}{3062500} a_{k +1}+\frac {2862406404}{765625} a_{k +2}-\frac {110139777}{1750} a_{k +3}+\frac {60042600949}{122500} a_{k +4}-\frac {1267229331}{700} a_{k +5}}{35 \left (1750 k^{2}+22747 k +73482\right )}, a_{1}=-\frac {47169 a_{0}}{85676500}, a_{2}=-\frac {589470647 a_{0}}{6299023500000}, a_{3}=-\frac {15579008605451309 a_{0}}{719673903835250000000}, a_{4}=-\frac {192276184430868189904453 a_{0}}{35585805046036360500000000000}, a_{5}=-\frac {214180356689737626731663904173563 a_{0}}{158618166712498345944341250000000000000}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +5 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +5\right )^{k +\frac {1747}{1750}}, a_{k +6}=-\frac {k^{2} a_{k}-39 k^{2} a_{k +1}+624 k^{2} a_{k +2}-5250 k^{2} a_{k +3}+24525 k^{2} a_{k +4}-60375 k^{2} a_{k +5}+\frac {872}{875} k a_{k}-\frac {102258}{875} k a_{k +1}+\frac {2728128}{875} k a_{k +2}-36732 k a_{k +3}+\frac {7722537}{35} k a_{k +4}-663954 k a_{k +5}-\frac {5241}{3062500} a_{k}-\frac {238261101}{3062500} a_{k +1}+\frac {2862406404}{765625} a_{k +2}-\frac {110139777}{1750} a_{k +3}+\frac {60042600949}{122500} a_{k +4}-\frac {1267229331}{700} a_{k +5}}{35 \left (1750 k^{2}+22747 k +73482\right )}, a_{1}=-\frac {47169 a_{0}}{85676500}, a_{2}=-\frac {589470647 a_{0}}{6299023500000}, a_{3}=-\frac {15579008605451309 a_{0}}{719673903835250000000}, a_{4}=-\frac {192276184430868189904453 a_{0}}{35585805046036360500000000000}, a_{5}=-\frac {214180356689737626731663904173563 a_{0}}{158618166712498345944341250000000000000}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +5\right )^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} \left (x +5\right )^{k +\frac {1747}{1750}}\right ), a_{k +6}=-\frac {k^{2} a_{k}-39 k^{2} a_{k +1}+624 k^{2} a_{k +2}-5250 k^{2} a_{k +3}+24525 k^{2} a_{k +4}-60375 k^{2} a_{k +5}-k a_{k}-39 k a_{k +1}+1872 k a_{k +2}-26250 k a_{k +3}+171678 k a_{k +4}-543411 k a_{k +5}+1248 a_{k +2}-31500 a_{k +3}+294319 a_{k +4}-1207680 a_{k +5}}{35 \left (1750 k^{2}+19253 k +52518\right )}, a_{1}=0, a_{2}=-\frac {a_{0}}{17530}, a_{3}=-\frac {20137 a_{0}}{1074632825}, a_{4}=-\frac {8578236977 a_{0}}{1580612944323000}, a_{5}=-\frac {233675437175659 a_{0}}{161423389882620381250}, b_{k +6}=-\frac {k^{2} b_{k}-39 k^{2} b_{k +1}+624 k^{2} b_{k +2}-5250 k^{2} b_{k +3}+24525 k^{2} b_{k +4}-60375 k^{2} b_{k +5}+\frac {872}{875} k b_{k}-\frac {102258}{875} k b_{k +1}+\frac {2728128}{875} k b_{k +2}-36732 k b_{k +3}+\frac {7722537}{35} k b_{k +4}-663954 k b_{k +5}-\frac {5241}{3062500} b_{k}-\frac {238261101}{3062500} b_{k +1}+\frac {2862406404}{765625} b_{k +2}-\frac {110139777}{1750} b_{k +3}+\frac {60042600949}{122500} b_{k +4}-\frac {1267229331}{700} b_{k +5}}{35 \left (1750 k^{2}+22747 k +73482\right )}, b_{1}=-\frac {47169 b_{0}}{85676500}, b_{2}=-\frac {589470647 b_{0}}{6299023500000}, b_{3}=-\frac {15579008605451309 b_{0}}{719673903835250000000}, b_{4}=-\frac {192276184430868189904453 b_{0}}{35585805046036360500000000000}, b_{5}=-\frac {214180356689737626731663904173563 b_{0}}{158618166712498345944341250000000000000}\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 -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying differential order: 2; exact nonlinear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> 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 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying to convert to an ODE of Bessel type trying to convert to an ODE of Bessel type -> trying reduction of order to Riccati trying Riccati sub-methods: -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3`[0, y]
✗ Solution by Maple
Order:=6; dsolve(x^3*(x^2-25)*(x-2)^2*diff(y(x),x$2)+3*x*(x-2)*diff(y(x),x)+7*(x+5)*y(x)=0,y(x),type='series',x=0);
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.092 (sec). Leaf size: 99
AsymptoticDSolveValue[x^3*(x^2-25)*(x-2)^2*y''[x]+3*x*(x-2)*y'[x]+7*(x+5)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (-\frac {1337698720169782190618881 x^5}{352638738432}+\frac {42840301537653264505 x^4}{3265173504}-\frac {344729362309955 x^3}{7558272}+\frac {3590248795 x^2}{23328}-\frac {50309 x}{108}+1\right ) x^{35/6}+\frac {c_1 e^{\left .\frac {3}{50}\right /x} \left (-\frac {37907198008560463448473952765642999 x^5}{5380840125000000000000000000}+\frac {27497874350326089989823180601 x^4}{7971615000000000000000}+\frac {10649898771731482781701 x^3}{14762250000000000}+\frac {975156065160301 x^2}{36450000000}+\frac {41066401 x}{135000}+1\right )}{x^{1159/300}} \]