56.32.7 problem Ex 7

Internal problem ID [14265]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 55. Summary. Page 129
Problem number : Ex 7
Date solved : Thursday, October 02, 2025 at 09:27:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 19

ode=(2*x^3-1)*D[y[x],{x,2}]-6*x^2*D[y[x],x]+6*x*y[x]==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: 0.342 (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) + (2*x**3 - 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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