60.4.23 problem 1479

Internal problem ID [11446]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1479
Date solved : Thursday, March 13, 2025 at 08:53:12 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 92

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

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

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 153

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

\[ y(x)\to \frac {1}{2} i c_2 x \, _1F_2\left (\frac {a}{2}+\frac {1}{2};\frac {3}{2},\frac {a}{2}+\frac {b}{2}+\frac {1}{2};\frac {x^2}{4}\right )+c_1 \, _1F_2\left (\frac {a}{2};\frac {1}{2},\frac {a}{2}+\frac {b}{2};\frac {x^2}{4}\right )+c_3 \left (\frac {i}{2}\right )^{-a-b+2} x^{-a-b+2} \, _1F_2\left (1-\frac {b}{2};-\frac {a}{2}-\frac {b}{2}+\frac {3}{2},-\frac {a}{2}-\frac {b}{2}+2;\frac {x^2}{4}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a*y(x) - x*Derivative(y(x), x) + x*Derivative(y(x), (x, 3)) + (a + b)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-a*y(x) + a*Derivative(y(x), (x, 2)) + b*Derivative(y(x), (x, 2)) + x*Derivative(y(x), (x, 3)))/x cannot be solved by the factorable group method