60.6.7 problem 1584

Internal problem ID [11544]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 5, linear fifth and higher order
Problem number : 1584
Date solved : Sunday, March 30, 2025 at 08:24:36 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 134

ode:=x*diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-m*n*diff(diff(diff(diff(y(x),x),x),x),x)+y(x)*a*x = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \frac {1}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_2 x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{5}, \frac {4}{5}, \frac {6}{5}, \frac {2}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_3 \,x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {4}{5}, \frac {6}{5}, \frac {7}{5}, \frac {3}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_4 \,x^{3} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {6}{5}, \frac {7}{5}, \frac {8}{5}, \frac {4}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_5 \,x^{m n +4} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {9}{5}+\frac {m n}{5}, \frac {8}{5}+\frac {m n}{5}, \frac {7}{5}+\frac {m n}{5}, \frac {6}{5}+\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right ) \]

✓ Mathematica. Time used: 1.204 (sec). Leaf size: 244

ode=x*D[y[x],{x,5}]-m*n*D[y[x],{x,4}]+a*x*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a*x*y(x) - m*n*Derivative(y(x), (x, 4)) + x*Derivative(y(x), (x, 5)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve a*x*y(x) - m*n*Derivative(y(x), (x, 4)) + x*Derivative(y(x), (x, 5))