53.1.69 problem 71

Internal problem ID [10541]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 71
Date solved : Tuesday, September 30, 2025 at 07:29:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y&=0 \end{align*}

✓ Maple. Time used: 0.291 (sec). Leaf size: 163

ode:=(3*x^2+x+1)*diff(diff(y(x),x),x)+(2+15*x)*diff(y(x),x)+12*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\frac {\sqrt {11}\, \arctan \left (\frac {\left (6 x +1\right ) \sqrt {11}}{11}\right )}{22}} \left (c_1 \left (i \sqrt {11}-6 x -1\right )^{{3}/{2}} \left (-36 x^{2}-12 x -12\right )^{-\frac {1}{4}+\frac {i \sqrt {11}}{44}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i \sqrt {11}}{22}, \frac {1}{2}+\frac {i \sqrt {11}}{22}\right ], \left [-\frac {1}{2}+\frac {i \sqrt {11}}{22}\right ], \frac {1}{2}+\frac {i \left (-6 x -1\right ) \sqrt {11}}{22}\right )+\left (i \sqrt {11}+6 x +1\right )^{\frac {5}{4}-\frac {i \sqrt {11}}{44}} \left (i \sqrt {11}-6 x -1\right )^{\frac {5}{4}+\frac {i \sqrt {11}}{44}} \operatorname {hypergeom}\left (\left [2, 2\right ], \left [\frac {5}{2}-\frac {i \sqrt {11}}{22}\right ], \frac {1}{2}+\frac {i \left (-6 x -1\right ) \sqrt {11}}{22}\right ) c_2 \right )}{\left (3 x^{2}+x +1\right )^{{5}/{4}}} \]

✓ Mathematica. Time used: 0.718 (sec). Leaf size: 113

ode=(1+x+3*x^2)*D[y[x],{x,2}]+(2+15*x)*D[y[x],x]+12*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x \exp \left (\int _1^x-\frac {3 K[1]}{6 K[1]^2+2 K[1]+2}dK[1]-\frac {1}{2} \int _1^x\frac {15 K[2]+2}{3 K[2]^2+K[2]+1}dK[2]\right ) \left (c_2 \int _1^x\frac {\exp \left (-2 \int _1^{K[3]}-\frac {3 K[1]}{6 K[1]^2+2 K[1]+2}dK[1]\right )}{K[3]^2}dK[3]+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((15*x + 2)*Derivative(y(x), x) + (3*x**2 + x + 1)*Derivative(y(x), (x, 2)) + 12*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False