Internal problem ID [9911]
Internal file name [OUTPUT/8858_Monday_June_06_2022_05_41_15_AM_28819537/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 5, linear fifth and higher order
Problem number: 1589.
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.
\[ \boxed {x^{\frac {5}{2}} y^{\left (5\right )}-a y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\frac {5}{2}} \left (\frac {d}{d x}\frac {d^{2}}{d x^{2}}y^{\prime \prime }\right )-a y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 5 \\ {} & {} & \frac {d}{d x}\frac {d^{2}}{d x^{2}}y^{\prime \prime } \end {array} \]
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: 486
dsolve(x^(2+1/2)*diff(y(x),x$5)-a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {8 \left (\left (x^{\frac {3}{2}} \left (\left (c_{2} -c_{5} \right ) \cos \left (\frac {\pi }{5}\right )+\left (-c_{3} +\frac {c_{4}}{15}\right ) \cos \left (\frac {2 \pi }{5}\right )+c_{1} \right ) a^{\frac {2}{5}}+\left (\frac {3 \left (c_{1} +\frac {c_{4}}{15}\right ) x \,a^{\frac {1}{5}}}{2}+\frac {3 \sqrt {x}\, \left (c_{2} +c_{3} \right )}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\left (\frac {3 x \left (c_{3} -c_{5} \right ) a^{\frac {1}{5}}}{2}+\frac {3 \sqrt {x}\, \left (c_{1} +c_{5} \right )}{4}\right ) \cos \left (\frac {2 \pi }{5}\right )+\frac {\sqrt {x}\, c_{4}}{20}+\frac {3 x c_{2} a^{\frac {1}{5}}}{2}\right ) \cos \left (2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}\right )+\sin \left (2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}\right ) \left (\left (\left (c_{2} +c_{5} \right ) \sin \left (\frac {\pi }{5}\right )+\sin \left (\frac {2 \pi }{5}\right ) \left (c_{3} +\frac {c_{4}}{15}\right )\right ) x^{\frac {3}{2}} a^{\frac {2}{5}}+\left (-\frac {3 \left (c_{1} -\frac {c_{4}}{15}\right ) x \,a^{\frac {1}{5}}}{2}-\frac {3 \sqrt {x}\, \left (c_{2} -c_{3} \right )}{4}\right ) \sin \left (\frac {\pi }{5}\right )-\frac {3 \sin \left (\frac {2 \pi }{5}\right ) \left (-2 x \left (c_{3} +c_{5} \right ) a^{\frac {1}{5}}+\sqrt {x}\, \left (c_{1} -c_{5} \right )\right )}{4}\right )\right ) {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}}+8 \left (\left (x^{\frac {3}{2}} \left (\left (c_{3} -\frac {c_{4}}{15}\right ) \cos \left (\frac {\pi }{5}\right )+\left (-c_{2} +c_{5} \right ) \cos \left (\frac {2 \pi }{5}\right )+c_{1} \right ) a^{\frac {2}{5}}+\left (-\frac {3 x \left (c_{3} -c_{5} \right ) a^{\frac {1}{5}}}{2}-\frac {3 \sqrt {x}\, \left (c_{1} +c_{5} \right )}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\left (-\frac {3 \left (c_{1} +\frac {c_{4}}{15}\right ) x \,a^{\frac {1}{5}}}{2}-\frac {3 \sqrt {x}\, \left (c_{2} +c_{3} \right )}{4}\right ) \cos \left (\frac {2 \pi }{5}\right )+\frac {\sqrt {x}\, c_{4}}{20}+\frac {3 x c_{2} a^{\frac {1}{5}}}{2}\right ) \cos \left (2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}\right )-\left (x^{\frac {3}{2}} \left (\left (c_{3} +\frac {c_{4}}{15}\right ) \sin \left (\frac {\pi }{5}\right )-\sin \left (\frac {2 \pi }{5}\right ) \left (c_{2} +c_{5} \right )\right ) a^{\frac {2}{5}}+\left (\frac {3 x \left (c_{3} +c_{5} \right ) a^{\frac {1}{5}}}{2}-\frac {3 \sqrt {x}\, \left (c_{1} -c_{5} \right )}{4}\right ) \sin \left (\frac {\pi }{5}\right )+\frac {3 \sin \left (\frac {2 \pi }{5}\right ) \left (2 \left (c_{1} -\frac {c_{4}}{15}\right ) x \,a^{\frac {1}{5}}+\sqrt {x}\, \left (c_{2} -c_{3} \right )\right )}{4}\right ) \sin \left (2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}\right )\right ) {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}}+4 \left (x^{\frac {3}{2}} a^{\frac {2}{5}}-\frac {3 a^{\frac {1}{5}} x}{2}+\frac {3 \sqrt {x}}{4}\right ) \left (c_{1} -c_{2} -c_{3} +\frac {c_{4}}{15}+c_{5} \right ) {\mathrm e}^{2 \sqrt {x}\, a^{\frac {1}{5}}}}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 206
DSolve[x^(2+1/2)*D[y[x],{x,5}]-a*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {4}{25} (-1)^{2/5} a^{2/5} c_2 x \, _0F_4\left (;-\frac {1}{5},\frac {1}{5},\frac {3}{5},\frac {7}{5};\frac {32 a x^{5/2}}{3125}\right )+\frac {16 \sqrt [5]{-1} a^{4/5} x^2 \left (625 (-1)^{3/5} c_3 \, _0F_4\left (;\frac {1}{5},\frac {3}{5},\frac {7}{5},\frac {9}{5};\frac {32 a x^{5/2}}{3125}\right )-4 a^{2/5} x \left (4 (-1)^{2/5} a^{2/5} c_5 x \, _0F_4\left (;\frac {7}{5},\frac {9}{5},\frac {11}{5},\frac {13}{5};\frac {32 a x^{5/2}}{3125}\right )+25 c_4 \, _0F_4\left (;\frac {3}{5},\frac {7}{5},\frac {9}{5},\frac {11}{5};\frac {32 a x^{5/2}}{3125}\right )\right )\right )}{390625}+c_1 \, _0F_4\left (;-\frac {3}{5},-\frac {1}{5},\frac {1}{5},\frac {3}{5};\frac {32 a x^{5/2}}{3125}\right ) \]