Internal problem ID [9159]
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.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
\[ \boxed {x y^{\left (5\right )}-m n y^{\prime \prime \prime \prime }+y a x=0} \]
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \frac {1}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{5}, \frac {4}{5}, \frac {6}{5}, \frac {2}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{3} x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {4}{5}, \frac {6}{5}, \frac {7}{5}, \frac {3}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{4} x^{3} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {6}{5}, \frac {7}{5}, \frac {8}{5}, \frac {4}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{5} x^{m n +4} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {9}{5}+\frac {m n}{5}, \frac {8}{5}+\frac {m n}{5}, \frac {7}{5}+\frac {m n}{5}, \frac {6}{5}+\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right ) \]
✓ Solution by Mathematica
Time used: 1.355 (sec). Leaf size: 244
DSolve[x*y'''''[x]-m*n*y''''[x]+a*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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 ) \\ \end{align*}