54.4.46 problem 1502

Internal problem ID [12768]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1502
Date solved : Friday, October 03, 2025 at 03:47:22 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime }-\left (x^{4}-6 x \right ) y^{\prime \prime }-\left (2 x^{3}-6\right ) y^{\prime }+2 x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.072 (sec). Leaf size: 103

ode:=x^2*diff(diff(diff(y(x),x),x),x)-(x^4-6*x)*diff(diff(y(x),x),x)-(2*x^3-6)*diff(y(x),x)+2*x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_3 \int {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (\operatorname {BesselK}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}-\operatorname {BesselK}\left (\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}-2 \operatorname {BesselK}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x +c_2 \int {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (\operatorname {BesselI}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}+\operatorname {BesselI}\left (-\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}-2 \operatorname {BesselI}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x +c_1}{x^{2}} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 72

ode=2*x^2*y[x] - (-6 + 2*x^3)*D[y[x],x] - (-6*x + x^4)*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 \frac {c_2 \int _1^x\operatorname {Hypergeometric1F1}\left (-\frac {2}{3},\frac {2}{3},\frac {K[1]^3}{3}\right )dK[1]+c_3 \int _1^x\sqrt [3]{-\frac {1}{3}} \operatorname {Hypergeometric1F1}\left (-\frac {1}{3},\frac {4}{3},\frac {K[2]^3}{3}\right ) K[2]dK[2]+c_1}{x^2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*y(x) + x**2*Derivative(y(x), (x, 3)) - (2*x**3 - 6)*Derivative(y(x), x) - (x**4 - 6*x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*(-x**3*Derivative(y(x), (x, 2)) + 2*x*y(x) + x*Derivative(y(x), (x, 3)) + 6*Derivative(y(x), (x, 2)))/(2*(x**3 - 3)) + Derivative(y(x), x) cannot be solved by the factorable group method