54.4.74 problem 1533

Internal problem ID [12796]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1533
Date solved : Friday, October 03, 2025 at 03:47:27 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-x y^{\prime }-n y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 58

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

\[ y = c_1 \operatorname {hypergeom}\left (\left [\frac {n}{3}\right ], \left [\frac {1}{3}, \frac {2}{3}\right ], \frac {x^{3}}{9}\right )+c_2 x \operatorname {hypergeom}\left (\left [\frac {1}{3}+\frac {n}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+c_3 \,x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}+\frac {n}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 106

ode=-(n*y[x]) - x*D[y[x],x] + Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{9} \left (3 \sqrt [3]{-3} c_2 x \, _1F_2\left (\frac {n}{3}+\frac {1}{3};\frac {2}{3},\frac {4}{3};\frac {x^3}{9}\right )+9 c_1 \, _1F_2\left (\frac {n}{3};\frac {1}{3},\frac {2}{3};\frac {x^3}{9}\right )+(-3)^{2/3} c_3 x^2 \, _1F_2\left (\frac {n}{3}+\frac {2}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{9}\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-n*y(x) - x*Derivative(y(x), x) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-n*y(x) + Derivative(y(x), (x, 3)))/x cannot be solved by the factorable group method