23.3.490 problem 496

Internal problem ID [6204]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 496
Date solved : Tuesday, September 30, 2025 at 02:36:17 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 6 x y+\left (-x^{3}+1\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 70

ode:=6*x*y(x)+(-x^3+1)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (2 x^{3}-2\right ) c_1 \sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{9}+\frac {c_1 \left (x^{3}-1\right ) \ln \left (x^{2}+x +1\right )}{9}+\frac {\left (-2 x^{3}+2\right ) c_1 \ln \left (x -1\right )}{9}+c_2 \,x^{3}-\frac {c_1 x}{3}-c_2 \]

✓ Mathematica. Time used: 0.065 (sec). Leaf size: 81

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

\begin{align*} y(x)&\to \frac {1}{9} c_2 \left (2 \sqrt {3} \left (x^3-1\right ) \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )-2 \left (x^3-1\right ) \log (1-x)-\log \left (x^2+x+1\right )+x^3 \log \left (x^2+x+1\right )-3 x\right )+c_1 \left (x^3-1\right ) \end{align*}

✓ Sympy. Time used: 0.150 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = \frac {\left (C_{1} \sqrt [3]{x^{3}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, 1 \\ \frac {4}{3} \end {matrix}\middle | {x^{3}} \right )} + C_{2} {{}_{1}F_{0}\left (\begin {matrix} -1 \\ \end {matrix}\middle | {x^{3}} \right )}\right ) \sqrt [3]{x^{3}}}{x} \]