54.4.16 problem 1469

Internal problem ID [12738]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1469
Date solved : Friday, October 03, 2025 at 03:47:16 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }+3 a x y^{\prime \prime }+3 a^{2} x^{2} y^{\prime }+a^{3} x^{3} y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 37

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

\[ y = {\mathrm e}^{-\frac {a \,x^{2}}{2}} \left (c_1 +c_2 \,{\mathrm e}^{\sqrt {3}\, \sqrt {a}\, x}+c_3 \,{\mathrm e}^{-\sqrt {3}\, \sqrt {a}\, x}\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 68

ode=a^3*x^3*y[x] + 3*a^2*x^2*D[y[x],x] + 3*a*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 e^{-\frac {a x^2}{2}-\sqrt {3} \sqrt {a} x} \left (c_1 e^{\sqrt {3} \sqrt {a} x}+c_3 e^{2 \sqrt {3} \sqrt {a} x}+c_2\right ) \end{align*}

✗ Sympy

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

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