59.1.128 problem 130

Internal problem ID [9300]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 130
Date solved : Sunday, March 30, 2025 at 02:31:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.068 (sec). Leaf size: 53

ode:=x^2*(2*x+1)*diff(diff(y(x),x),x)+x*(3*x^2+14*x+5)*diff(y(x),x)+(12*x^2+18*x+4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\frac {3 x}{2}} \left (\left (1+2 x \right )^{{1}/{4}} \operatorname {HeunC}\left (-\frac {3}{4}, \frac {1}{4}, 0, \frac {21}{32}, -\frac {5}{32}, 1+2 x \right ) c_2 +\operatorname {HeunC}\left (-\frac {3}{4}, -\frac {1}{4}, 0, \frac {21}{32}, -\frac {5}{32}, 1+2 x \right ) c_1 \right )}{\left (1+2 x \right )^{{1}/{4}} x^{2}} \]

✓ Mathematica. Time used: 0.337 (sec). Leaf size: 120

ode=x^2*(1+2*x)*D[y[x],{x,2}]+x*(5+14*x+3*x^2)*D[y[x],x]+(4+18*x+12*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(2*x + 1)*Derivative(y(x), (x, 2)) + x*(3*x**2 + 14*x + 5)*Derivative(y(x), x) + (12*x**2 + 18*x + 4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False