54.3.369 problem 1386

Internal problem ID [12664]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1386
Date solved : Wednesday, October 01, 2025 at 02:19:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 58

ode:=diff(diff(y(x),x),x) = 18/(2*x+1)^2/(x^2+x+1)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {36 \left (x^{2}+x +1\right ) c_2 \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )-16 c_2 \left (x^{3}+x^{2}+\frac {11}{8} x +\frac {3}{16}\right ) \sqrt {3}+c_1 \left (x^{2}+x +1\right )}{\left (2 x +1\right )^{2}} \]

✓ Mathematica. Time used: 0.098 (sec). Leaf size: 80

ode=D[y[x],{x,2}] == (18*y[x])/((1 + 2*x)^2*(1 + x + x^2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - 18*y(x)/((2*x + 1)**2*(x**2 + x + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), (x, 2)) - 18*y(x)/((2*x + 1)**2*(x**2 + x + 1)) cannot be solved by the hypergeometric method