54.4.3 problem 1451

Internal problem ID [12725]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1451
Date solved : Friday, October 03, 2025 at 03:47:15 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-a \,x^{b} y&=0 \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 114

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 164

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

\begin{align*} y(x)&\to (-1)^{\frac {1}{b+3}} (b+3)^{-\frac {6}{b+3}} x a^{\frac {1}{b+3}} \left ((-1)^{\frac {1}{b+3}} c_3 x a^{\frac {1}{b+3}} \, _0F_2\left (;1+\frac {1}{b+3},1+\frac {2}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right )+(b+3)^{\frac {3}{b+3}} c_2 \, _0F_2\left (;1-\frac {1}{b+3},1+\frac {1}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right )\right )+c_1 \, _0F_2\left (;1-\frac {2}{b+3},1-\frac {1}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right ) \end{align*}

✗ Sympy

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

NotImplementedError : solve: Cannot solve -a*x**b*y(x) + Derivative(y(x), (x, 3))