23.5.111 problem 111

Internal problem ID [6720]
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 : 111
Date solved : Friday, October 03, 2025 at 02:09:49 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} -12 y+3 \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (x^{2}+1\right ) y^{\prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 60

ode:=-12*y(x)+3*(2*x^2+1)*diff(diff(y(x),x),x)+x*(x^2+1)*diff(diff(diff(y(x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.182 (sec). Leaf size: 69

ode=-12*y[x] + 3*(1 + 2*x^2)*D[y[x],{x,2}] + x*(1 + x^2)*D[y[x],{x,3}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{6} \left (-3 c_3 x \sqrt {x^2+1} \text {arctanh}\left (\sqrt {x^2+1}\right )+c_1 \left (4 x^2+2\right )+2 c_2 x \sqrt {x^2+1}+3 c_3 x+\frac {c_3}{x}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2 + 1)*Derivative(y(x), (x, 3)) + (6*x**2 + 3)*Derivative(y(x), (x, 2)) - 12*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve x*(x**2 + 1)*Derivative(y(x), (x, 3)) + (6*x**2 + 3)*Deriv