54.4.15 problem 1468

Internal problem ID [12737]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1468
Date solved : Friday, October 03, 2025 at 03:47:16 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-6 x y^{\prime \prime }+2 \left (4 x^{2}+2 a -1\right ) y^{\prime }-8 y a x&=0 \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 59

ode:=diff(diff(diff(y(x),x),x),x)-6*x*diff(diff(y(x),x),x)+2*(4*x^2+2*a-1)*diff(y(x),x)-8*y(x)*a*x = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 57

ode=-8*a*x*y[x] + 2*(-1 + 2*a + 4*x^2)*D[y[x],x] - 6*x*D[y[x],{x,2}] + Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \operatorname {HermiteH}\left (\frac {a}{2},x\right ) \operatorname {Hypergeometric1F1}\left (-\frac {a}{4},\frac {1}{2},x^2\right )+c_1 \operatorname {HermiteH}\left (\frac {a}{2},x\right )^2+c_3 \operatorname {Hypergeometric1F1}\left (-\frac {a}{4},\frac {1}{2},x^2\right )^2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-8*a*x*y(x) - 6*x*Derivative(y(x), (x, 2)) + (4*a + 8*x**2 - 2)*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) - (8*a*x*y(x) + 6*x*Derivative(y(x), (x, 2)) - Derivative(y(x), (x, 3)))/(2*(2*a + 4*x**2 - 1)) cannot be solved by the factorable group method