6.7 problem 1584

Internal problem ID [9906]
Internal file name [OUTPUT/8853_Monday_June_06_2022_05_40_43_AM_86438534/index.tex]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 5, linear fifth and higher order
Problem number: 1584.
ODE order: 5.
ODE degree: 1.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[[_high_order, _with_linear_symmetries]]

Unable to solve or complete the solution.

\[ \boxed {x y^{\left (5\right )}-m n y^{\prime \prime \prime \prime }+a x y=0} \] Unable to solve this ODE.

6.7.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}\frac {d^{2}}{d x^{2}}y^{\prime \prime }\right ) x -m n \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right )+a x y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 5 \\ {} & {} & \frac {d}{d x}\frac {d^{2}}{d x^{2}}y^{\prime \prime } \\ \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 {m n}{x}, P_{3}\left (x \right )=0, P_{4}\left (x \right )=0, P_{5}\left (x \right )=0, P_{6}\left (x \right )=a \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}}}=-m n \\ {} & \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}}}=0 \\ {} & \circ & x^{3}\cdot P_{4}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{3}\cdot P_{4}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=0 \\ {} & \circ & x^{4}\cdot P_{5}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{4}\cdot P_{5}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=0 \\ {} & \circ & x^{5}\cdot P_{6}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{5}\cdot P_{6}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=0 \\ {} & \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 {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 \cdot y\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -1 \\ {} & {} & x \cdot y=\moverset {\infty }{\munderset {k =1}{\sum }}a_{k -1} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) \left (k +r -2\right ) \left (k +r -3\right ) x^{k +r -4} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime }=\moverset {\infty }{\munderset {k =-4}{\sum }}a_{k +4} \left (k +4+r \right ) \left (k +3+r \right ) \left (k +2+r \right ) \left (k +r +1\right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x \cdot \left (\frac {d}{d x}\frac {d^{2}}{d x^{2}}y^{\prime \prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot \left (\frac {d}{d x}\frac {d^{2}}{d x^{2}}y^{\prime \prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) \left (k +r -2\right ) \left (k +r -3\right ) \left (k +r -4\right ) x^{k +r -4} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +4 \\ {} & {} & x \cdot \left (\frac {d}{d x}\frac {d^{2}}{d x^{2}}y^{\prime \prime }\right )=\moverset {\infty }{\munderset {k =-4}{\sum }}a_{k +4} \left (k +4+r \right ) \left (k +3+r \right ) \left (k +2+r \right ) \left (k +r +1\right ) \left (k +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} r \left (-1+r \right ) \left (-2+r \right ) \left (-3+r \right ) \left (-m n +r -4\right ) x^{-4+r}+a_{1} \left (1+r \right ) r \left (-1+r \right ) \left (-2+r \right ) \left (-m n +r -3\right ) x^{-3+r}+a_{2} \left (2+r \right ) \left (1+r \right ) r \left (-1+r \right ) \left (-m n +r -2\right ) x^{-2+r}+a_{3} \left (3+r \right ) \left (2+r \right ) \left (1+r \right ) r \left (-m n +r -1\right ) x^{-1+r}+a_{4} \left (4+r \right ) \left (3+r \right ) \left (2+r \right ) \left (1+r \right ) \left (-m n +r \right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k +4} \left (k +4+r \right ) \left (k +3+r \right ) \left (k +2+r \right ) \left (k +r +1\right ) \left (-m n +k +r \right )+a a_{k -1}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & r \left (-1+r \right ) \left (-2+r \right ) \left (-3+r \right ) \left (-m n +r -4\right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, 1, 2, 3, m n +4\right \} \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} x \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [a_{1} \left (1+r \right ) r \left (-1+r \right ) \left (-2+r \right ) \left (-m n +r -3\right )=0, a_{2} \left (2+r \right ) \left (1+r \right ) r \left (-1+r \right ) \left (-m n +r -2\right )=0, a_{3} \left (3+r \right ) \left (2+r \right ) \left (1+r \right ) r \left (-m n +r -1\right )=0, a_{4} \left (4+r \right ) \left (3+r \right ) \left (2+r \right ) \left (1+r \right ) \left (-m n +r \right )=0\right ] \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & a_{k +4} \left (k +4+r \right ) \left (k +3+r \right ) \left (k +2+r \right ) \left (k +r +1\right ) \left (-m n +k +r \right )+a a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & a_{k +5} \left (k +5+r \right ) \left (k +4+r \right ) \left (k +3+r \right ) \left (k +2+r \right ) \left (-m n +k +r +1\right )+a a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +5}=-\frac {a a_{k}}{\left (k +5+r \right ) \left (k +4+r \right ) \left (k +3+r \right ) \left (k +2+r \right ) \left (-m n +k +r +1\right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +5}=-\frac {a a_{k}}{\left (k +5\right ) \left (k +4\right ) \left (k +3\right ) \left (k +2\right ) \left (-m n +k +1\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +5}=-\frac {a a_{k}}{\left (k +5\right ) \left (k +4\right ) \left (k +3\right ) \left (k +2\right ) \left (-m n +k +1\right )}, 0=0, 0=0, 0=0, -24 a_{4} m n =0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =1 \\ {} & {} & a_{k +5}=-\frac {a a_{k}}{\left (k +6\right ) \left (k +5\right ) \left (k +4\right ) \left (k +3\right ) \left (-m n +k +2\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +1}, a_{k +5}=-\frac {a a_{k}}{\left (k +6\right ) \left (k +5\right ) \left (k +4\right ) \left (k +3\right ) \left (-m n +k +2\right )}, 0=0, 0=0, -24 a_{3} m n =0, 120 a_{4} \left (-m n +1\right )=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =2 \\ {} & {} & a_{k +5}=-\frac {a a_{k}}{\left (k +7\right ) \left (k +6\right ) \left (k +5\right ) \left (k +4\right ) \left (-m n +k +3\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =2 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +2}, a_{k +5}=-\frac {a a_{k}}{\left (k +7\right ) \left (k +6\right ) \left (k +5\right ) \left (k +4\right ) \left (-m n +k +3\right )}, 0=0, -24 a_{2} m n =0, 120 a_{3} \left (-m n +1\right )=0, 360 a_{4} \left (-m n +2\right )=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =3 \\ {} & {} & a_{k +5}=-\frac {a a_{k}}{\left (k +8\right ) \left (k +7\right ) \left (k +6\right ) \left (k +5\right ) \left (-m n +k +4\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =3 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +3}, a_{k +5}=-\frac {a a_{k}}{\left (k +8\right ) \left (k +7\right ) \left (k +6\right ) \left (k +5\right ) \left (-m n +k +4\right )}, -24 a_{1} m n =0, 120 a_{2} \left (-m n +1\right )=0, 360 a_{3} \left (-m n +2\right )=0, 840 a_{4} \left (-m n +3\right )=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =m n +4 \\ {} & {} & a_{k +5}=-\frac {a a_{k}}{\left (m n +k +9\right ) \left (m n +k +8\right ) \left (m n +k +7\right ) \left (m n +k +6\right ) \left (k +5\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =m n +4 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{m n +k +4}, a_{k +5}=-\frac {a a_{k}}{\left (m n +k +9\right ) \left (m n +k +8\right ) \left (m n +k +7\right ) \left (m n +k +6\right ) \left (k +5\right )}, a_{1} \left (m n +5\right ) \left (m n +4\right ) \left (m n +3\right ) \left (m n +2\right )=0, 2 a_{2} \left (m n +6\right ) \left (m n +5\right ) \left (m n +4\right ) \left (m n +3\right )=0, 3 a_{3} \left (m n +7\right ) \left (m n +6\right ) \left (m n +5\right ) \left (m n +4\right )=0, 4 a_{4} \left (m n +8\right ) \left (m n +7\right ) \left (m n +6\right ) \left (m n +5\right )=0\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} x^{k +1}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} x^{k +2}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}e_{k} x^{k +3}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}f_{k} x^{m n +k +4}\right ), b_{k +5}=-\frac {a b_{k}}{\left (k +5\right ) \left (k +4\right ) \left (k +3\right ) \left (k +2\right ) \left (-m n +k +1\right )}, 0=0, 0=0, 0=0, -24 b_{4} m n =0, c_{k +5}=-\frac {a c_{k}}{\left (k +6\right ) \left (k +5\right ) \left (k +4\right ) \left (k +3\right ) \left (-m n +k +2\right )}, 0=0, 0=0, -24 c_{3} m n =0, 120 c_{4} \left (-m n +1\right )=0, d_{k +5}=-\frac {a d_{k}}{\left (k +7\right ) \left (k +6\right ) \left (k +5\right ) \left (k +4\right ) \left (-m n +k +3\right )}, 0=0, -24 d_{2} m n =0, 120 d_{3} \left (-m n +1\right )=0, 360 d_{4} \left (-m n +2\right )=0, e_{k +5}=-\frac {a e_{k}}{\left (k +8\right ) \left (k +7\right ) \left (k +6\right ) \left (k +5\right ) \left (-m n +k +4\right )}, -24 e_{1} m n =0, 120 e_{2} \left (-m n +1\right )=0, 360 e_{3} \left (-m n +2\right )=0, 840 e_{4} \left (-m n +3\right )=0, f_{k +5}=-\frac {a f_{k}}{\left (m n +k +9\right ) \left (m n +k +8\right ) \left (m n +k +7\right ) \left (m n +k +6\right ) \left (k +5\right )}, f_{1} \left (m n +5\right ) \left (m n +4\right ) \left (m n +3\right ) \left (m n +2\right )=0, 2 f_{2} \left (m n +6\right ) \left (m n +5\right ) \left (m n +4\right ) \left (m n +3\right )=0, 3 f_{3} \left (m n +7\right ) \left (m n +6\right ) \left (m n +5\right ) \left (m n +4\right )=0, 4 f_{4} \left (m n +8\right ) \left (m n +7\right ) \left (m n +6\right ) \left (m n +5\right )=0\right ] \end {array} \]

Maple trace

`Methods for high order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying high order exact linear fully integrable 
trying to convert to a linear ODE with constant coefficients 
trying differential order: 5; missing the dependent variable 
trying a solution in terms of MeijerG functions 
<- MeijerG function solution successful`
 

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 134

dsolve(x*diff(y(x),x$5)-m*n*diff(y(x),x$4)+a*x*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \frac {1}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{5}, \frac {4}{5}, \frac {6}{5}, \frac {2}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{3} x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {4}{5}, \frac {6}{5}, \frac {7}{5}, \frac {3}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{4} x^{3} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {6}{5}, \frac {7}{5}, \frac {8}{5}, \frac {4}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{5} x^{m n +4} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {6}{5}+\frac {m n}{5}, \frac {9}{5}+\frac {m n}{5}, \frac {8}{5}+\frac {m n}{5}, \frac {7}{5}+\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right ) \]

✓ Solution by Mathematica

Time used: 2.144 (sec). Leaf size: 244

DSolve[x*y'''''[x]-m*n*y''''[x]+a*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{625} x \left (x \left (5 a^{3/5} c_4 x \, _0F_4\left (;\frac {6}{5},\frac {7}{5},\frac {8}{5},\frac {4}{5}-\frac {m n}{5};-\frac {a x^5}{3125}\right )+25 a^{2/5} c_3 \, _0F_4\left (;\frac {4}{5},\frac {6}{5},\frac {7}{5},\frac {3}{5}-\frac {m n}{5};-\frac {a x^5}{3125}\right )+c_5 5^{-m n} a^{\frac {1}{5} (m n+4)} x^{m n+2} \, _0F_4\left (;\frac {m n}{5}+\frac {6}{5},\frac {m n}{5}+\frac {7}{5},\frac {m n}{5}+\frac {8}{5},\frac {m n}{5}+\frac {9}{5};-\frac {a x^5}{3125}\right )\right )+125 \sqrt [5]{a} c_2 \, _0F_4\left (;\frac {3}{5},\frac {4}{5},\frac {6}{5},\frac {2}{5}-\frac {m n}{5};-\frac {a x^5}{3125}\right )\right )+c_1 \, _0F_4\left (;\frac {2}{5},\frac {3}{5},\frac {4}{5},\frac {1}{5}-\frac {m n}{5};-\frac {a x^5}{3125}\right ) \]