23.3.519 problem 525

Internal problem ID [6233]
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 : 525
Date solved : Tuesday, September 30, 2025 at 02:37:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -6 x y+6 x^{2} y^{\prime }+\left (-2 x^{3}+1\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 14

ode:=-6*x*y(x)+6*x^2*diff(y(x),x)+(-2*x^3+1)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \,x^{3}+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.077 (sec). Leaf size: 19

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

\begin{align*} y(x)&\to c_1 x-c_2 \left (x^3+1\right ) \end{align*}

✓ Sympy. Time used: 0.333 (sec). Leaf size: 82

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

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