76.50.1 problem Ex. 3

Internal problem ID [20292]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at page 113
Problem number : Ex. 3
Date solved : Thursday, October 02, 2025 at 05:41:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} y^{\prime \prime }+\left (-6 x^{2}+2\right ) y^{\prime }-4 y&=0 \end{align*}

✓ Maple. Time used: 0.069 (sec). Leaf size: 107

ode:=3*x^2*diff(diff(y(x),x),x)+(-6*x^2+2)*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {x}\, \left ({\mathrm e}^{2 x} \operatorname {HeunD}\left (-\frac {8 \sqrt {3}}{3}, -1-\frac {8 \sqrt {3}}{3}, \frac {16 \sqrt {3}}{3}, -\frac {8 \sqrt {3}}{3}+1, \frac {\sqrt {3}\, x -1}{\sqrt {3}\, x +1}\right ) c_2 +{\mathrm e}^{\frac {2}{3 x}} \operatorname {HeunD}\left (\frac {8 \sqrt {3}}{3}, -1-\frac {8 \sqrt {3}}{3}, \frac {16 \sqrt {3}}{3}, -\frac {8 \sqrt {3}}{3}+1, \frac {\sqrt {3}\, x -1}{\sqrt {3}\, x +1}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.187 (sec). Leaf size: 39

ode=3*x^2*D[y[x],{x,2}]+(2-6*x^2)*D[y[x],x]-4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{2 x} \left (c_2 \int _1^xe^{\frac {2}{3 K[1]}-2 K[1]}dK[1]+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*Derivative(y(x), (x, 2)) + (2 - 6*x**2)*Derivative(y(x), x) - 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (3*x**2*Derivative(y(x), (x, 2)) - 4*y(x))/(2*(3*x**2 - 1)) cannot be solved by the factorable group method