59.1.68 problem 70

Internal problem ID [9240]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 70
Date solved : Wednesday, March 05, 2025 at 07:45:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 62

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

\[ y = \frac {\left (x +\frac {1}{3}\right ) \left (x -6\right ) c_{1} {\mathrm e}^{-\frac {x}{3}} \left (\Gamma \left (-\frac {1}{9}\right )+\frac {10 \Gamma \left (-\frac {10}{9}, -\frac {x}{3}-\frac {1}{9}\right )}{9}\right ) \left (-\frac {x}{3}-\frac {1}{9}\right )^{{1}/{9}}}{9}+3 c_{2} \left (x +\frac {1}{3}\right ) \left (x -6\right ) {\mathrm e}^{-\frac {x}{3}} \left (\frac {x}{3}+\frac {1}{9}\right )^{{1}/{9}}-\frac {10 c_{1} {\mathrm e}^{\frac {1}{9}}}{9} \]

✓ Mathematica. Time used: 0.421 (sec). Leaf size: 103

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

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

✗ Sympy

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

False