60.6.11 problem 1588

Internal problem ID [11587]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 5, linear fifth and higher order
Problem number : 1588
Date solved : Tuesday, January 28, 2025 at 06:06:57 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{10} y^{\left (5\right )}-a y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.028 (sec). Leaf size: 90

dsolve(x^10*diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-a*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {6}{5}, \frac {7}{5}, \frac {8}{5}, \frac {9}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {4}{5}, \frac {6}{5}, \frac {7}{5}, \frac {8}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_3 \,x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{5}, \frac {4}{5}, \frac {6}{5}, \frac {7}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_4 \,x^{3} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \frac {6}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_5 \,x^{4} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{5}, \frac {2}{5}, \frac {3}{5}, \frac {4}{5}\right ], -\frac {a}{3125 x^{5}}\right ) \]

✓ Solution by Mathematica

Time used: 6.095 (sec). Leaf size: 106

DSolve[x^10*D[y[x],{x,5}]-a*y[x]== 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {x^4 \left (c_1 e^{-\frac {\sqrt [5]{a}}{x}}+c_2 e^{\frac {\sqrt [5]{-1} \sqrt [5]{a}}{x}}+c_3 e^{-\frac {(-1)^{2/5} \sqrt [5]{a}}{x}}+c_4 e^{\frac {(-1)^{3/5} \sqrt [5]{a}}{x}}+c_5 e^{-\frac {(-1)^{4/5} \sqrt [5]{a}}{x}}\right )}{e^4} \]