54.4.41 problem 1497

Internal problem ID [12763]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1497
Date solved : Friday, October 03, 2025 at 03:47:20 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime }-3 \left (p +q \right ) x y^{\prime \prime }+3 p \left (3 q +1\right ) y^{\prime }-x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 77

ode:=x^2*diff(diff(diff(y(x),x),x),x)-3*(p+q)*x*diff(diff(y(x),x),x)+3*p*(3*q+1)*diff(y(x),x)-x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {hypergeom}\left (\left [\right ], \left [-p +\frac {2}{3}, -q +\frac {1}{3}\right ], \frac {x^{3}}{27}\right )+c_2 \,x^{3 p +1} \operatorname {hypergeom}\left (\left [\right ], \left [p +\frac {4}{3}, \frac {2}{3}-q +p \right ], \frac {x^{3}}{27}\right )+c_3 \,x^{3 q +2} \operatorname {hypergeom}\left (\left [\right ], \left [q +\frac {5}{3}, \frac {4}{3}+q -p \right ], \frac {x^{3}}{27}\right ) \]

✓ Mathematica. Time used: 0.133 (sec). Leaf size: 127

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

\begin{align*} y(x)&\to c_1 \, _0F_2\left (;\frac {2}{3}-p,\frac {1}{3}-q;\frac {x^3}{27}\right )+c_2 (-1)^{p+\frac {1}{3}} 3^{-3 p-1} x^{3 p+1} \, _0F_2\left (;p+\frac {4}{3},p-q+\frac {2}{3};\frac {x^3}{27}\right )+c_3 (-1)^{q+\frac {2}{3}} 3^{-3 q-2} x^{3 q+2} \, _0F_2\left (;q+\frac {5}{3},-p+q+\frac {4}{3};\frac {x^3}{27}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
p = symbols("p") 
q = symbols("q") 
y = Function("y") 
ode = Eq(3*p*(3*q + 1)*Derivative(y(x), x) - x**2*y(x) + x**2*Derivative(y(x), (x, 3)) - x*(3*p + 3*q)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - x*(3*p*Derivative(y(x), (x, 2)) + 3*q*Derivative(y(x), (x, 2)) + x*y(x) - x*Derivative(y(x), (x, 3)))/(3*p*(3*q + 1)) cannot be solved by the factorable group method