53.1.484 problem 500

Internal problem ID [10956]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 500
Date solved : Tuesday, September 30, 2025 at 07:33:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{5}+1\right ) y^{\prime \prime }+14 x^{4} y^{\prime }+10 x^{3} y&=0 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 30

ode:=(2*x^5+1)*diff(diff(y(x),x),x)+14*x^4*diff(y(x),x)+10*x^3*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 x}{\left (2 x^{5}+1\right )^{{2}/{5}}}+c_2 \operatorname {hypergeom}\left (\left [\frac {1}{5}, 1\right ], \left [\frac {4}{5}\right ], -2 x^{5}\right ) \]

✓ Mathematica. Time used: 0.641 (sec). Leaf size: 41

ode=(1+2*x^5)*D[y[x],{x,2}]+14*x^4*D[y[x],x]+10*x^3*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1 x-c_2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{5},\frac {3}{5},\frac {4}{5},-2 x^5\right )}{\left (2 x^5+1\right )^{2/5}} \end{align*}

✓ Sympy. Time used: 0.342 (sec). Leaf size: 87

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(14*x**4*Derivative(y(x), x) + 10*x**3*y(x) + (2*x**5 + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt [5]{2 x^{5} + 1} \left (C_{1} \sqrt [5]{\frac {x^{5}}{2 x^{5} + 1}} {{}_{2}F_{1}\left (\begin {matrix} 0, \frac {2}{5} \\ \frac {6}{5} \end {matrix}\middle | {\frac {2 x^{5}}{2 x^{5} + 1}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{5}, \frac {1}{5} \\ \frac {4}{5} \end {matrix}\middle | {\frac {2 x^{5}}{2 x^{5} + 1}} \right )}\right ) \left (x^{5}\right )^{\frac {2}{5}}}{x^{2} \left (x^{5} + \frac {1}{2}\right )^{\frac {2}{5}}} \]