53.1.85 problem 87

Internal problem ID [10557]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 87
Date solved : Tuesday, September 30, 2025 at 07:29:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+x \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.284 (sec). Leaf size: 162

ode:=x^2*(10*x^2+x+5)*diff(diff(y(x),x),x)+x*(48*x^2+3*x+4)*diff(y(x),x)+(36*x^2+x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (i \sqrt {199}+20 x +1\right )^{-\frac {i \sqrt {199}}{1990}} \left (i \sqrt {199}-20 x -1\right )^{\frac {i \sqrt {199}}{1990}} {\mathrm e}^{-\frac {\sqrt {199}\, \arctan \left (\frac {\left (20 x +1\right ) \sqrt {199}}{199}\right )}{995}} \left (x^{{1}/{5}} \operatorname {HeunG}\left (\frac {\sqrt {199}+i}{i-\sqrt {199}}, 0, 0, \frac {1}{5}, \frac {6}{5}, -\frac {i \sqrt {199}}{995}, -\frac {20 x}{1+i \sqrt {199}}\right ) c_2 +\operatorname {HeunG}\left (\frac {\sqrt {199}+i}{i-\sqrt {199}}, \frac {15721-179 i \sqrt {199}}{194275 i \sqrt {199}+641775}, -\frac {1}{5}, 0, \frac {4}{5}, -\frac {i \sqrt {199}}{995}, -\frac {20 x}{1+i \sqrt {199}}\right ) c_1 \right )}{10 x^{2}+x +5} \]

✓ Mathematica. Time used: 0.738 (sec). Leaf size: 132

ode=x^2*(5+x+10*x^2)*D[y[x],{x,2}]+x*(4+3*x+48*x^2)*D[y[x],x]+(x+36*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \exp \left (\int _1^x\left (\frac {3}{5 K[1]}-\frac {1}{10 \left (10 K[1]^2+K[1]+5\right )}\right )dK[1]-\frac {1}{2} \int _1^x\frac {48 K[2]^2+3 K[2]+4}{10 K[2]^3+K[2]^2+5 K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\left (\frac {3}{5 K[1]}-\frac {1}{10 \left (10 K[1]^2+K[1]+5\right )}\right )dK[1]\right )dK[3]+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(10*x**2 + x + 5)*Derivative(y(x), (x, 2)) + x*(48*x**2 + 3*x + 4)*Derivative(y(x), x) + (36*x**2 + x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False