23.5.62 problem 62

Internal problem ID [6671]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 62
Date solved : Friday, October 03, 2025 at 02:09:38 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

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

ode:=a^3*x^3*y(x)+3*a^2*x^2*diff(y(x),x)+3*a*x*diff(diff(y(x),x),x)+diff(diff(diff(y(x),x),x),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: 0.005 (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}] + D[y[x],{x,3}] == 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