53.1.136 problem 138

Internal problem ID [10608]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 138
Date solved : Tuesday, September 30, 2025 at 07:30:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (5+9 x \right ) y^{\prime }+\left (4+3 x \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 73

ode:=x^2*(2*x+1)*diff(diff(y(x),x),x)+x*(5+9*x)*diff(y(x),x)+(3*x+4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_2 \left (\frac {1}{2}+x \right )^{2} \ln \left (\sqrt {1+2 x}-1\right )-c_2 \left (\frac {1}{2}+x \right )^{2} \ln \left (\sqrt {1+2 x}+1\right )+c_2 \left (x +\frac {2}{3}\right ) \sqrt {1+2 x}+4 c_1 \left (\frac {1}{2}+x \right )^{2}}{x^{2} \sqrt {1+2 x}} \]

✓ Mathematica. Time used: 0.136 (sec). Leaf size: 110

ode=x^2*(1+2*x)*D[y[x],{x,2}]+x*(5+9*x)*D[y[x],x]+(4+3*x)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \exp \left (\int _1^x\frac {7 K[1]+1}{4 K[1]^2+2 K[1]}dK[1]-\frac {1}{2} \int _1^x\frac {9 K[2]+5}{2 K[2]^2+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {7 K[1]+1}{4 K[1]^2+2 K[1]}dK[1]\right )dK[3]+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(2*x + 1)*Derivative(y(x), (x, 2)) + x*(9*x + 5)*Derivative(y(x), x) + (3*x + 4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False