23.5.108 problem 108

Internal problem ID [6717]
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 : 108
Date solved : Tuesday, September 30, 2025 at 03:51:05 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} -\left (-a \,x^{3}+12\right ) y+6 x^{2} y^{\prime \prime }+x^{3} y^{\prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 135

ode:=-(-a*x^3+12)*y(x)+6*x^2*diff(diff(y(x),x),x)+x^3*diff(diff(diff(y(x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\left (\left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{{2}/{3}}+i a^{3} x \right ) c_2 \,{\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2 a}}-c_3 \left (\left (-i-\sqrt {3}\right ) \left (-a^{4}\right )^{{2}/{3}}+i a^{3} x \right ) {\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2 a}}+c_1 \left (a^{3} x +2 \left (-a^{4}\right )^{{2}/{3}}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} x}{a}}}{x^{3}} \]

✓ Mathematica. Time used: 0.215 (sec). Leaf size: 97

ode=-((12 - a*x^3)*y[x]) + 6*x^2*D[y[x],{x,2}] + x^3*D[y[x],{x,3}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1 e^{-\sqrt [3]{a} x} \left (\sqrt [3]{a} x+2\right )+c_2 e^{\sqrt [3]{-1} \sqrt [3]{a} x} \left (\sqrt [3]{a} x+2 (-1)^{2/3}\right )+c_3 e^{-(-1)^{2/3} \sqrt [3]{a} x} \left (\sqrt [3]{a} x-2 \sqrt [3]{-1}\right )}{x^3} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 3)) + 6*x**2*Derivative(y(x), (x, 2)) + (a*x**3 - 12)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve x**3*Derivative(y(x), (x, 3)) + 6*x**2*Derivative(y(x), (x