59.1.550 problem 566

Internal problem ID [9722]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 566
Date solved : Sunday, March 30, 2025 at 02:45:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (-x^{2}+11 x +5\right ) y^{\prime }+\left (-7 x^{2}+16 x +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.131 (sec). Leaf size: 36

ode:=9*x^2*(1+x)*diff(diff(y(x),x),x)+3*x*(-x^2+11*x+5)*diff(y(x),x)+(-7*x^2+16*x+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\frac {c_1 \operatorname {HeunC}\left (-\frac {1}{3}, -\frac {4}{3}, 0, -\frac {1}{9}, \frac {11}{18}, 1+x \right )}{\left (1+x \right )^{{4}/{3}}}+c_2 \operatorname {HeunC}\left (-\frac {1}{3}, \frac {4}{3}, 0, -\frac {1}{9}, \frac {11}{18}, 1+x \right )}{x^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.33 (sec). Leaf size: 120

ode=9*x^2*(1+x)*D[y[x],{x,2}]+3*x*(5+11*x-x^2)*D[y[x],x]+(1+16*x-7*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {K[1]^2+3 K[1]+3}{6 K[1]^2+6 K[1]}dK[1]-\frac {1}{2} \int _1^x\frac {1}{3} \left (\frac {7}{K[2]+1}-1+\frac {5}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]^2+3 K[1]+3}{6 K[1]^2+6 K[1]}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*(x + 1)*Derivative(y(x), (x, 2)) + 3*x*(-x**2 + 11*x + 5)*Derivative(y(x), x) + (-7*x**2 + 16*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False