54.4.7 problem 1455

Internal problem ID [12729]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1455
Date solved : Friday, October 03, 2025 at 03:47:15 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+\left (a +b -1\right ) x y^{\prime }-b y a&=0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 71

ode:=diff(diff(diff(y(x),x),x),x)-x^2*diff(diff(y(x),x),x)+(b-1+a)*x*diff(y(x),x)-b*y(x)*a = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 127

ode=-(a*b*y[x]) + (-1 + a + b)*x*D[y[x],x] - x^2*D[y[x],{x,2}] + Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt [3]{-\frac {1}{3}} c_2 x \, _2F_2\left (\frac {1}{3}-\frac {a}{3},\frac {1}{3}-\frac {b}{3};\frac {2}{3},\frac {4}{3};\frac {x^3}{3}\right )+c_1 \, _2F_2\left (-\frac {a}{3},-\frac {b}{3};\frac {1}{3},\frac {2}{3};\frac {x^3}{3}\right )+\left (-\frac {1}{3}\right )^{2/3} c_3 x^2 \, _2F_2\left (\frac {2}{3}-\frac {a}{3},\frac {2}{3}-\frac {b}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{3}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a*b*y(x) - x**2*Derivative(y(x), (x, 2)) + x*(a + b - 1)*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) - (a*b*y(x) + x**2*Derivative(y(x), (x, 2)) - Derivative(y(x), (x, 3)))/(x*(a + b - 1)) cannot be solved by the factorable group method